偏序度量空间中多值压缩的耦合重合

摘要

本文研究了偏序度量空间中多值非线性压缩的耦合重合点的存在性。我们从两种不同的方法来做这件事，第一种是最近在Samet和Vetro（偏序度量空间中多值非线性压缩映象的耦合不动点定理，非线性Anal）中研究的Δ-对称性。74，4260-4268（2011）），第二个是混合的克-Lakshmikantham和cci-irić研究的单调性（偏序度量空间中非线性压缩的耦合不动点定理，非线性Anal。70, 4341-4349 (2009)).

所提出的定理推广了由于多值非线性压缩映象的某些结果。71，2716-2723（2009）），Samet和Vetro（偏序度量空间中多值非线性压缩映射的耦合不动点定理，非线性分析。744260-4268（2011））和其他许多人。我们通过建立一个示例来支持结果。

2000 MSC：初级06F30；46B20；47E10。

1.简介和序言

让(X（X）,d日)是一个度量空间。我们表示为CB（断路器）(X（X）)非空闭有界子集的集合X（X）。对于A类,B类 ∈ CB（断路器）(X（X）)和x个 ∈ X（X），假设是这样

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}}{<trans\; data-src="inf">inf公司</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="and">和</trans>}}\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=",">,</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="max">最大值</trans><trans\; data-src="\{">\{</trans>\underset{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}}{<trans\; data-src="sup">啜饮</trans>}\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\underset{\mathrm{<trans\; data-src="b">b</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="B">B类</trans>}}{<trans\; data-src="sup">啜饮</trans>}\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="b">b</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src=".">.</trans>$$

这样的映射H（H）被称为Hausdorff度量CB（断路器）(X（X）)诱发因素d。

定义1.1.元素x ∈ X被称为多值映射T的不动点以下为：X（X）→CB（断路器）(X（X）)当且仅当x ∈ 发送。

1969年，纳德勒[1]将著名的Banach压缩原理从单值映射推广到多值映射，并证明了以下多值压缩的不动点定理。

定理1.1.让(X（X）,d日)是一个完备的度量空间，设T是从X到CB的映射(X（X）).假设存在c ∈[0,1）这样H(Tx（发送）,泰伊) ≤光盘(x个,年)对于所有x，y ∈ 那么，T有一个不动点。

许多作者在不同的条件下研究了各种多值压缩映射不动点的存在性。1989年，沟口和高桥[2]证明了以下关于弱压缩的有趣的不动点定理。

定理1.2.让(十、 d日)是一个完备的度量空间，设T是从X到CB的映射(X（X）).假设H(Tx（发送）,泰伊) ≤α(d日(x个,年))d日(x个,年)对于所有x,年 ∈ 十、 其中α是来自的函数[0,∞)到[0,1)满足条件$<trans\; data-src="lim">林</trans>{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="+">+</trans>}}\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$对于所有t ∈[0, ∞).那么，T有一个不动点。

让$\mathrm{<trans\; data-src="C">C类</trans>}\mathrm{<trans\; data-src="L">L（左）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\Phi ">\Phi </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="Ā">Ā</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\}">\}</trans>$，其中$\mathrm{<trans\; data-src="Ā">Ā</trans>}$表示的闭合A类在度量空间中(X（X）,d日). 在这种情况下，奇·伊里奇[三]证明了以下有趣的定理。

定理1.3.（请参见[三])让(十、 d日)是一个完备度量空间，设T是从X到CL的映射(X（X）).让f以下为：X（X）→ℝ 是f定义的函数(x个) =d日(x个,发送)对于所有x ∈ X.假设f是下半连续的，并且存在函数: [0, +∞) → [一, 1), 0 <一< 1,令人满意的

$$\underset{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(1.1)

假设对于任何x ∈ X有y ∈ Tx满足以下两个条件：

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$$

(1.2)

使得

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(1.3)

那么，存在z ∈ X使z ∈ Tz公司。

定义1.2. [4]设X是非空集，F以下为：X（X）×X→X是一个给定的映射。一个元素(x个,年)∈ X（X）×如果F，则称X为映射F的耦合不动点(x、 年) =x和F(年,x个) =年.

定义1.3. [5]让(x个,年)∈ X（X）×X（X）,F类以下为：X（X）×X（X）→X和g以下为：X（X）→十、我们这么说(x、 年)是F和g的耦合重合点，如果F(x个,年) =gx和F(年,x个) =x，y的gy ∈ X（X）.

定义1.4.A函数f以下为：X（X）×X（X）→ℝ 称为下半连续当且仅当对于任何序列{x个 _n个 }⊂ X（X）, {年 _n个 }⊂ X和(x个,年)∈ X（X）×X、 我们有

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Rightarrow ">\Rightarrow </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="liminf">liminf公司</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

让(X（X）,d日)是具有偏序的度量空间G公司以下为：X（X）→X（X）是一个给定的映射。我们定义集合Δ⊂ X（X）×X（X）通过

$$\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\preccurlyeq ">\preccurlyeq </trans>\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src=".">.</trans>$$

在[6]Samet和Vetro引入了二进制关系R（右）在氯(X（X）)由定义

$$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="R">R（右）</trans>}\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="\iff ">\iff </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="\subseteq ">\subseteq </trans>\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=",">,</trans>$$

哪里A类,B类 ∈ 氯(X（X）).

定义1.5.让F以下为：X（X）×X（X）→氯(X（X）)是一个给定的映射。我们说F是一个Δ-对称映射当且仅当(x个,年)∈Δ⇒ F类(x个,年)射频(年,x个).

示例1.1.假设X=[0,1]，被赋予通常的秩序≤.让G: [0,1] → [0,1]是G定义的映射(x个) =M代表所有x ∈[0,1],其中M是中的常数[0,1].然后，Δ=[0,1]×[0,1]F是一个Δ-对称映射。

定义1.6. [6]让F以下为：X（X）×X（X）→氯(X（X）)是一个给定的映射。我们这么说(x个,年)∈ X（X）×X是F的耦合不动点当且仅当X ∈ F类(x、 年)和y ∈ F类(年,x个).

定义1.7.让F以下为：X（X）×X（X）→氯(X（X）)是一个给定的映射，设g以下为：X（X）→十、我们这么说(x个,年)∈ X（X）×X是F和g的耦合重合点当且仅当gx ∈ F类(x个,年)和gy ∈ F类(年,x个).

在[6]Samet和Vetro证明了定理1.3的以下耦合不动点版本。

定理1.4.让(X（X）,d日)是具有偏序的完备度量空间 ≼.我们假设$\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$,即存在(x个₀,年₀)∈Δ.设F以下为：X（X）×X（X）→氯(X（X）)成为Δ-对称映射。假设函数f以下为：X（X）×X（X）→ [0,+∞)由定义

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$$

是下半连续的，并且存在函数: [0, ∞) → [一, 1), 0 <一< 1,令人满意的

$$\underset{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

假设对于任何(x个,年)∈Δ存在u ∈ F类(x、 年)和v ∈ F类(年,x个)令人满意的

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$$

使得

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

那么，F承认一个耦合不动点，即存在z= (z（z）₁,z（z）₂)∈ X（X）×X使z₁ ∈ F类(z（z）₁,z（z）₂)和z₂ ∈ F类(z（z）₂,z（z）₁).

2006年，Bhaskar和Lakshmikantham[4]引入了耦合不动点的概念，并在偏序度量空间中建立了一些耦合不动点定理。他们讨论了周期边值问题解的存在唯一性。拉克希米坎塔姆和奇·伊里奇[5]用混合方法证明了偏序完备度量空间中非线性压缩映象的耦合重合和耦合公共不动点定理克-单调性。有关耦合不动点理论的更多详细信息，请参阅[7–12]以及其中的参考文献。在这里，我们使用两种不同的方法研究了多值非线性压缩的耦合重合的存在性，第一种方法是基于最近在[6]第二种是基于混合克-Lakshmikantham和cci-irić研究的单调性[5]. 所提出的定理扩展了由于cirć而得到的某些结果[三]萨梅特和维特罗[6]以及许多其他方面。我们通过建立一个示例来支持结果。

2.Δ对称性耦合重合

以下是本节的主要结果，它概括了上述关于chc irić、Samet和Vetro的结果。

定理2.1.让(X（X）,d日)是具有偏序的度量空间 ≼ 和$\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$.假设F以下为：X（X）×X（X）→氯(X（X）)是一个Δ-对称映射，g以下为：X（X）→X是连续的，gX是完全的，函数f以下为：克(X（X）) ×克(X（X）) → [0, +∞)由定义

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$$

是下半连续的，并且存在函数: [0, ∞) → [一, 1), 0 <一< 1,令人满意的

$$\underset{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(2.1)

假设对于任何(x个,年)∈Δ存在gu ∈ F类(x个,年)和gv ∈ F类(年,x个)令人满意的

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$$

（2.2）

使得

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

(2.3)

那么，F和g有一个耦合的重合点，即存在gz= (gz（高斯）₁,gz（高斯）₂)∈ X（X）×X表示gz₁ ∈ F类(z（z）₁,z（z）₂)和gz₂ ∈ F类(z（z）₂,z（z）₁).

证明.根据定义ϕ我们有ϕ(如果(x个,年))每个<1(x个,年)∈ X（X）×X（X），对于任何(x个,年)∈ X（X）×X（X）存在古 ∈ F类(x个,年)和全球价值观 ∈ F类(年,x个)这样的话

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

和

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

因此，对于每个(x个,年)∈ X（X）×X（X），存在古 ∈ F类(x个,年)和全球价值观 ∈ F类(年,x个)令人满意（2.2）。

让(x个₀,年₀)∈Δ是任意的和固定的。通过（2.2）和（2.3），我们可以选择盖克斯₁ ∈ F类(x个₀,年₀)和吉₁ ∈ F类(年₀,x个₀)这样的话

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans>$$

(2.4)

和

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

(2.5)

从（2.4）和（2.5）中，我们可以得到

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>& <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\\ <trans\; data-src="=">=</trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="\{">\{</trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\}">\}</trans>\\ <trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>\end{array}$$

因此，

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(2.6)

现在，因为F类是Δ-对称映射(x个₀,年₀)∈Δ，我们有

$$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="R">R（右）</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\Rightarrow ">\Rightarrow </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=".">.</trans>$$

此外，通过（2.2）和（2.3），我们可以选择盖克斯₂ ∈ F类(x个₁,年₁)和吉₂ ∈ F类(年₁,x个₁)这样的话

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>$$

和

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

因此，我们得到

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

与(x个₂,年₂)∈Δ.

继续这个过程，我们可以选择{盖克斯 _n个 }⊂ X（X）和{吉 _n个 }⊂ X（X）这样所有人n个 ∈ ℕ，我们有

$$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2.7)

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2.8)

和

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(2.9)

现在，我们来展示一下如果(盖克斯 _n个 ,吉 _n个 ) → 0作为n个→ ∞. 我们假设如果(盖克斯 _n个 ,吉 _n个 )全部>0n个 ∈ ℕ，因为如果如果(盖克斯 _n个 ,吉 _n个 )某些情况下为0n个 ∈ ℕ，然后我们得到D类(盖克斯 _n个 ,F类(x个 _n个 ,年 _n个 ))=0，这意味着$\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$和D类(吉 _n个 ,F类(年 _n个 ,x个 _n个 ))=0，这意味着吉 _n个 ∈ F类(年 _n个 ,x个 _n个 ). 因此，在这种情况下(x个 _n个 ,年 _n个 )是一个耦合的重合点F类和克并证明了定理的断言。

从（2.9）和ϕ(t吨)<1，我们推断{如果(盖克斯 _n个 ,吉 _n个 )}是正实数的严格递减序列。因此，有一些δ≥0，使得

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=".">.</trans>$$

现在，我们将证明这一点δ= 0. 假设情况并非如此；考虑到（2.9）两边的限制，并考虑到假设（2.1），我们有

$$\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=",">,</trans>$$

矛盾。因此，δ=0，即，

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

(2.10)

现在，让我们证明一下{盖克斯 _n个 }和{吉 _n个 }柯西序列在(X（X）,d日). 假设

$$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="0">0</trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src=".">.</trans>$$

然后，根据假设（2.1），我们得到α< 1. 让q个是这样的α<q个< 1. 那么，有一些n个₀ ∈ ℕ使得

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="q">q个</trans>}\mathsf{\text{<trans data-src="for">对于</trans>}}\mathsf{\text{<trans data-src="each">每个</trans>}}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

因此，从（2.9）中，我们得到

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="q">q个</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="for">对于</trans>}}\mathsf{\text{<trans data-src="each">每个</trans>}}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

因此通过诱导，

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="q">q个</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="for">对于</trans>}}\mathsf{\text{<trans data-src="each">每个</trans>}}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.11)

自ϕ(t吨) ≥一>0代表全部t吨≥0，从（2.8）和（2.11），我们得到

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="a">一</trans>}}}{\mathrm{<trans\; data-src="q">q个</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="for">对于</trans>}}\mathsf{\text{<trans data-src="each">每个</trans>}}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.12)

自（2.12）起q个<1，我们的结论是{盖克斯 _n个 }和{吉 _n个 }柯西序列在(X（X）,d日).

现在，因为gX公司已完成，有一个w个= (w个₁,w个₂)∈ gX公司×gX公司使得

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{\text{<trans data-src="and">和</trans>}}\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$$

(2.13)

对一些人来说z（z）₁,z（z）₂在里面X（X）。我们现在展示一下z（z）= (z（z）₁,z（z）₂)是一个耦合的重合点F类和克.根据假设如果是下半连续的，所以从（2.10）可以得到

$$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="liminf">黎明</trans>}\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

因此，

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

这意味着gz（高斯）₁ ∈ F类(z（z）₁,z（z）₂)和gz（高斯）₂ ∈ F类(z（z）₂,z（z）₁)即。，z（z）= (z（z）₁,z（z）₂)是一个耦合的重合点F类和克。这就完成了证明。

现在，我们证明以下定理。

定理2.2.让(X（X）,d日)是具有偏序的度量空间 ≼ 和$\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$.假设F以下为：X（X）×X→氯(X（X）)是一个Δ-对称映射，g以下为：X（X）→X（X）是连续且gX完成。假设函数f以下为：gX公司×gX公司→ [0，+∞）定理2.1中定义的是下半连续的，并且存在函数: [0, +∞) → [一, 1), 0 <一< 1,令人满意的

$$\underset{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>}{<trans\; data-src="limsup">酸橙酱</trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(2.14)

假设对于任何(x个,年)∈Δ,存在gu ∈ F类(x个,年)和gv ∈ F类(年,x个)令人满意的

$$\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

(2.15)

使得

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

(2.16)

那么，F和g有一个耦合的重合点，即存在z= (z（z）₁,z（z）₂)∈ X（X）×X表示gz₁ ∈ F类(z（z）₁,z（z）₂)和gz₂ ∈ F类(z（z）₂,z（z）₁).

证明.更换ϕ(如果(x、 年))带有ϕ(d日(盖克斯,古) +d日(吉,全球价值观))按照定理2.1的证明，可以构造迭代序列{x个 _n个 }⊂ X（X）和{年 _n个 }⊂ X（X）这样所有人n个 ∈ ℕ，我们有

$$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2.17)

$$\begin{array}{c}\sqrt{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\\ <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\end{array}$$

(2.18)

和

$$\begin{array}{c}\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\end{array}$$

(2.19)

为所有人n个≥ 0. 再次，根据定理2.1的证明，我们得出如下结论{D类(盖克斯 _n个 ,F类(x个 _n个 ,年 _n个 )) +D类(吉 _n个 ,F类(年 _n个 ,x个 _n个 ))}是正实数的严格递减序列。因此，有一些δ≥0，使得

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=".">.</trans>$$

(2.20)

因为在我们的假设中ϕ(d日(盖克斯 _n个 ,盖克斯_n个+1) +d日(吉 _n个 ,吉_n个+1))，我们需要证明{d日(盖克斯 _n个 ,盖克斯_n个+1) +d日(吉 _n个 ,吉_n个+1)}允许一个子序列收敛到某个η⁺对一些人来说η≥ 0. 自φ(t吨) ≥一>对于所有t≥0的0，从（2.18）我们得到

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="a">一</trans>}}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

(2.21)

根据（2.20）和（2.21），我们得出如下结论：{d日(盖克斯 _n个 ,盖克斯_n个+1)+d日(吉 _n个 ,吉_n个+1)}有界。因此，有一些θ≥0，以便

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="liminf">liminf公司</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src=".">.</trans>$$

(2.22)

自盖克斯_n个+1∈ F类(x个 _n个 ,年 _n个 )和吉_n个+1∈ F类(年 _n个 ,x个 _n个 )，因此

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

对于每个n个≥ 0. 这意味着θ≥δ现在，我们将展示θ=δ。如果我们假设δ=0，那么从（2.20）和（2.21）我们得到

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

因此，如果δ=0，则θ=δ假设现在δ>相反，假设θ>δ.然后，θ-δ>（2.20）和（2.22）中有一个正整数n个₀使得

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}$$

(2.23)

和

$$\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>$$

(2.24)

为所有人n个≥n个₀然后，结合（2.18）、（2.23）和（2.24），我们得到

$$\begin{array}{c}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\left(\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\\ <trans\; data-src="<"><</trans>\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\\ <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\end{array}$$

为所有人n个≥n个₀因此，我们得到

$$\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="3">三</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="3">三</trans>}\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}$$

（2.25）

为所有人n个≥n个₀.设置$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="3">三</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}}{\mathrm{<trans\; data-src="3">三</trans>}\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$现在，从（2.19）和（2.25）可以看出

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$$

为所有人n个≥n个₀最后，因为我们假设δ>0和as小时<1，通过归纳并结合上述不等式，得出如下结论

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="\delta ">\delta </trans>}& <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="h">小时</trans>}}^{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\end{array}$$

对于正整数k个₀，这与假设相矛盾θ>δ所以我们必须θ=δ现在，我们将展示θ= 0. 自

$$\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

所以我们可以把（2.22）读作

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="liminf">liminf公司</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\theta ">\theta </trans>}}^{<trans\; data-src="+">+</trans>}<trans\; data-src=".">.</trans>$$

因此，存在子序列$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$使得

$$\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\theta ">\theta </trans>}}^{<trans\; data-src="+">+</trans>}<trans\; data-src=".">.</trans>$$

现在，到（2.14），我们已经

$$\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="\theta ">\theta </trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

(2.26)

从（2.19），

$$\begin{array}{c}\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="\le ">\le </trans>\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>\end{array}$$

将限额视为k个→ +∞ 使用（2.20），我们得到

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="\delta ">\delta </trans>}& <trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>\\ <trans\; data-src="\le ">\le </trans>\left(\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\right)\\ <trans\; data-src="(">(</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="=">=</trans>\left(\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="\theta ">\theta </trans>}}^{<trans\; data-src="+">+</trans>}}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}\right)\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=".">.</trans>\end{array}$$

根据上一个不等式，如果我们假设δ>0，我们得到

$$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="+">+</trans>}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>$$

与（2.26）的矛盾。因此，δ= 0. 然后，从（2.20）和（2.21）我们得到

$$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="+">+</trans>}{<trans\; data-src="limsup">酸橙酱</trans>}\sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

再一次，按照定理2.1的证明进行，可以证明{盖克斯 _n个 }和{吉 _n个 }柯西序列在gX公司还有那个z（z）= (z（z）₁,z（z）₂)∈ X（X）×X（X）是一个耦合的重合点F类,克，即。

$$\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="and">和</trans>}}\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

示例2.3。假设X（X）=[0,1]，配备常用公制d日以下为：X（X）×X（X）→ [0，+∞），以及G公司: [0,1] → [0,1]是由定义的映射

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}\mathsf{\text{<trans data-src="for all ">为所有人</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>$$

哪里M（M）是[0,1]中的常量。让F类以下为：X（X）×X（X）→氯(X（X）)定义为

$$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\hfill \frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\hfill \\ \hfill <trans\; data-src="\{">\{</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="96">96</trans>}}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="5">5</trans>}}<trans\; data-src="\}">\}</trans>\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

然后，Δ=[0,1]×[0,1]和F类是一个Δ-对称映射。立即定义φ: [0, +∞) → [0,1）由

$$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\hfill \frac{\mathrm{<trans\; data-src="11">11</trans>}}{\mathrm{<trans\; data-src="12">12</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\hfill \\ \hfill \frac{\mathrm{<trans\; data-src="11">11</trans>}}{\mathrm{<trans\; data-src="18">18</trans>}}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

让克: [0,1] → [0,1]定义为盖克斯=x个²现在，我们将展示F类(x个,年)满足定理2.2的所有假设。让

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\sqrt{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src="+">+</trans>\sqrt{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\hfill & \text{<trans data-src="if ">如果</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\hfill \\ \sqrt{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="160">160</trans>}}\hfill & \text{<trans data-src="if ">如果</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>\text{<trans data-src=" and ">以及</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\hfill \\ \sqrt{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="160">160</trans>}}\hfill & \text{<trans data-src="if ">如果</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\text{<trans data-src="[0,">[0,</trans>}\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>\text{<trans data-src=" and ">以及</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\hfill \\ \frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="80">80</trans>}}\hfill & \text{<trans data-src="if ">如果</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

很容易看出该函数

$$\mathrm{<trans\; data-src="f">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\hfill \mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\hfill \\ \hfill \mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="160">160</trans>}}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>\mathsf{\text{<trans data-src=" and ">以及</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\hfill \\ \hfill \mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="160">160</trans>}}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>\mathsf{\text{<trans data-src=" and ">以及</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\hfill \\ \hfill \frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="80">80</trans>}}\hfill & \hfill \mathsf{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}\hfill \end{array}\right.$$

是下半连续的。因此，对于所有人来说x个,年 ∈[0,1]带$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\ne ">\ne </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}$，存在$\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="\}">\}</trans>$和$\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="\}">\}</trans>$使得

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="4">4</trans>}}}{\mathrm{<trans\; data-src="64">64</trans>}}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="4">4</trans>}}}{\mathrm{<trans\; data-src="64">64</trans>}}\\ <trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\left[\left(\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\left(\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)<trans\; data-src="+">+</trans>\left(\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\left(\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\right]\\ <trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\left[\left(\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\left(\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans>\right]\\ <trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="max">最大值</trans>\left\{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="+">+</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right\}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\\ <trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="11">11</trans>}}{\mathrm{<trans\; data-src="12">12</trans>}}<trans\; data-src="max">最大值</trans>\left\{\left(\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)<trans\; data-src=",">,</trans>\left(\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="4">4</trans>}}\right)\right\}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\\ <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>\end{array}$$

因此，对于x，y∈[0,1]带$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\ne ">\ne </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}$，满足条件（2.15）和（2.16）。根据类似的论据，可以很容易地证明条件（2.15）和（2.16）也满足$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\cup ">\cup </trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$和$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}$最后，对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="32">32</trans>}}$如果我们假设$\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="96">96</trans>}}$，因此$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}$.

因此，我们得到

$$\begin{array}{ll}\hfill \sqrt{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>& <trans\; data-src="=">=</trans>\sqrt{\frac{\mathrm{<trans\; data-src="11">11</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}<trans\; data-src="\cdot ">\cdot </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}}<trans\; data-src="\cdot ">\cdot </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}\\ <trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="43">43</trans>}}{\mathrm{<trans\; data-src="80">80</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\end{array}$$

和

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}\left|\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="96">96</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}{\left(\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="96">96</trans>}}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}\right|\\ <trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="11">11</trans>}}{\mathrm{<trans\; data-src="12">12</trans>}}<trans\; data-src="\cdot ">\cdot </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}<trans\; data-src="\cdot ">\cdot </trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{\mathrm{<trans\; data-src="24">24</trans>}}\\ <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>\\ \hfill \text{<trans data-src="(4) ">(4) </trans>}\end{array}$$

因此，我们得出结论，定理2.2的所有条件都满足，并且F类,克承认一个耦合的巧合点z（z）= (0, 0).

3.混合的耦合巧合克-单调性

最近，在偏序度量空间中不动点的存在性领域有了令人兴奋的发展（参见[13–24]). 使用交换映射和混合的概念克-单调性，Lakshmikantham和cci-irić[5]建立了耦合重合点结果的存在性，推广了Bhaskar和Lakshmikantham的结果[4]. Choudhury和Kundu将这些结果推广到兼容的地图。在本节中，我们将扩展交换、兼容映射和混合的概念克-当F类是多值映射，并证明了上述结果的推广。

具有混合单调性质的类似项Lakshmikantham和ch-irić[5]介绍了以下混合的概念克-单调性。

定义3.1.让(X（X）,≼)是偏序集且F以下为：X（X）×X（X）→X和g以下为：X（X）→X（X）.我们说F类有混合g-单调性：如果F在其第一个参数中是单调g-非递减的，并且在其第二个参数中也是单调g-不递增的，即对于任何x，y ∈ X（X）,

$${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\preccurlyeq ">\preccurlyeq </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="p">第页</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\preccurlyeq ">\preccurlyeq </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$$

(3.1)

和

$${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\preccurlyeq ">\preccurlyeq </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="p">第页</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\succcurlyeq ">\succcurlyeq </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(3.2)

定义3.2.让(X（X）,≼)是偏序集，F以下为：X（X）×X（X）→氯(X（X）)让g以下为：X（X）→X是一个映射。我们说映射F具有混合g-单调性，如果，对于所有x₁,x个₂,年₁,年₂ ∈ X（X）具有盖克斯₁ ≼ 盖克斯₂和gy₁ ≽ 吉₂,我们得到了所有的gu₁ ∈ F类(x个₁,年₁)存在gu₂ ∈ F类(x个₂,年₂)这样gu₁ ≼ 古₂对于所有gv₁ ∈ F类(年₁,x个₁)存在gv₂ ∈ F类(年₂,x个₂)这样gv₁ ≽ z重力₂.

定义3.3.映射F以下为：X（X）×X（X）→CB（断路器）(X（X）)和g以下为：X（X）→X表示兼容，如果

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

和

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

无论何时{x个 _n个 }和{年 _n个 }是X中的序列，因此X=极限_n个→∞盖克斯 _n个 ∈林_n个→∞F类(x个 _n个 ,年 _n个 )和y=极限_n个→∞吉 _n个 ∈林_n个→∞F类(年 _n个 ,x个 _n个 ),对于所有x，y ∈ X表示满意.

定义3.4.映射F以下为：X（X）×X（X）→CB（断路器）(X（X）)和g以下为：X（X）→如果gF，则称X为通勤(x个,年)⊆ F类(盖克斯,吉)对于所有x，y ∈ X（X）.

引理3.1. [1]如果是A，B ∈ CB（断路器）(X（X）)带H(A类,B类) < ϵ,然后针对每个a ∈ A存在元素b ∈ B使d(一,b) < ϵ.

引理3.2. [1]让{A类 _n个 }是CB中的序列(X（X）)和林_n个→∞H（H）(A类 _n个 ,A类) = 0对于A ∈ CB（断路器）(X（X）).如果x _n个 ∈ A类 _n个 和林_n个→∞d日(x个 _n个 ,x个) = 0,然后是x ∈ A类.

让(X（X）,≼)是部分有序集，并且d日是上的度量X（X）这样的话(X（X）,d日)是一个完整的度量空间。我们定义了乘积空间上的偏序X（X）×X（X）作为：

的(u个,v（v）),(x个,年)∈ X（X）×X（X）, (u个,v（v）)≼(x个,年)当且仅当u个 ≼ x个,v（v） ≽ 年.

上的产品度量X（X）×X（X）定义为

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathsf{\text{<trans data-src="for all ">为所有人</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

为了方便记法，我们使用了相同的符号d日对于产品度量以及上的度量X（X）.

我们从以下结果开始，该结果给出了相容映射的耦合重合点的存在性F类和克在偏序度量空间中，其中F类是多值映射。

定理3.1.让F以下为：X（X）×X（X）→CB（断路器）(X（X）),克以下为：X（X）→X应：

• (1) 存在κ ∈(0,1)具有

  $$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">如果</trans>}\mathrm{<trans\; data-src="o">哦</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="l">我</trans>}\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\succcurlyeq ">\succcurlyeq </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=";">;</trans>$$

• (2) 如果gx ₁ ≼ 盖克斯 ₂,吉 ₂ ≼ 吉 ₁,x个 _我 ,年 _我 ∈ X（X）(我= 1,2),那么对于所有gu ₁ ∈ F类(x个 ₁,年 ₁)存在gu ₂ ∈ F类(x个 ₂,年 ₂)和gu一起 ₁ ≼ 古 ₂ 对于所有gv ₁ ∈ F类(年 ₁,x个 ₁)存在gv ₂ ∈ F类(年 ₂,x个 ₂)带有gv ₂ ≼ 全球价值观 ₁ 提供d((古 ₁,全球价值观 ₁), (古 ₂,全球价值观 ₂)) < 1;即F具有混合g-单调性质，前提是d((古 ₁,全球价值观 ₁), (古 ₂,全球价值观 ₂))<1；

• (3) 那里存在x ₀,年 ₀ ∈ 十、 还有一些gx ₁ ∈ F类(x个 ₀,年 ₀),吉 ₁ ∈ F类(年 ₀,x个 ₀)带gx ₀ ≼ 盖克斯 ₁,吉 ₀ ≽ 吉 ₁ 这样d((盖克斯 ₀,吉 ₀), (盖克斯 ₁,吉 ₁)) < 1 -κ、 其中κ ∈(0,1);

• (4) 如果是非递减序列{x个 _n个 }→x、 然后是x _n个 ≤x表示所有n，如果是非递增序列{年 _n个 }→y、 然后是y≤年 _n个 对于所有n和gX都是完整的.

那么，F和g有一个耦合的重合点.

证明.让x个₀,年₀ ∈ X（X）然后通过（3）存在盖克斯₁ ∈ F类(x个₀,年₀),吉₁ ∈ F类(年₀,x个₀)带有盖克斯₀ ≼ 盖克斯₁,吉₀ ≽ 吉₁使得

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=".">.</trans>$$

(3.3)

自(盖克斯₀,吉₀)≼(盖克斯₁,吉₁)使用（1）和（3.3），我们有

$$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>$$

和类似的

$$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

利用（2）和引理3.1，我们得到了盖克斯₂ ∈ F类(x个₁,年₁),吉₂ ∈ F类(年₁,x个₁)带有x个₁ ≼ x个₂和年₁ ≽ 年₂使得

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>$$

(3.4)

和

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(3.5)

从（3.4）和（3.5），

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(3.6)

再次通过（1）和（3.6），我们得到

$$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>$$

和

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

从引理3.1和（2），我们得到了盖克斯_三 ∈ F类(x个₂,年₂),吉_三 ∈ F类(年₂,x个₂)带有盖克斯₂ ≼ 盖克斯_三,吉₂ ≽ 吉_三使得

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>$$

和

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

由此可见

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

继续这样，我们获得盖克斯_n个+1∈ F类(x个 _n个 ,年 _n个 ),吉_n个+1∈ F类(年 _n个 ,x个 _n个 )带有$\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\preccurlyeq ">\preccurlyeq </trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\succcurlyeq ">\succcurlyeq </trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}}$使得

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>$$

和

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

因此，

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(3.7)

接下来，我们将展示{盖克斯 _n个 }是中的Cauchy序列X（X）.让米>n个.然后，

$$\begin{array}{ll}\hfill \mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>& <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="]">]</trans>\frac{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="]">]</trans>\frac{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}\right]\frac{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="=">=</trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\frac{{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\\ \hfill \text{<trans data-src="(6) ">(6) </trans>}\end{array}$$

因为κ ∈（0,1），1-κ^m-n个< 1. 因此，d日(盖克斯 _n个 ,盖克斯 _米 ) → 0作为n个→ ∞ 意味着{盖克斯 _n个 }是一个柯西序列。同样，我们可以证明{吉 _n个 }也是Cauchy序列X（X）.自gX公司已完成，存在x个,年 ∈ X（X）使得盖克斯 _n个 →盖克斯和吉 _n个 →吉作为n个→ ∞. 最后，我们必须证明这一点盖克斯 ∈ F类(x个,年)和吉 ∈ F类(年,x个).

自{盖克斯 _n个 }是一个非递减序列，并且{吉 _n个 }是中的非递增序列X（X）使得盖克斯 _n个 →x个和吉 _n个 →年作为n个→ ∞, 因此我们有盖克斯 _n个 ≼ x个和吉 _n个 ≽ 年为所有人n个.作为n个→ ∞, （1） 意味着

$$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

自盖克斯_n个+1∈ F类(x个 _n个 ,年 _n个 )和lim_n个→∞d日(盖克斯_n个+1,盖克斯)=0，然后使用引理3.2盖克斯 ∈ F类(x个,年). 再次通过（1），

$$\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{\mathrm{<trans\; data-src="\kappa ">\kappa </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

自吉_n个+1∈ F类(年 _n个 ,x个 _n个 )和lim_n个→∞d日(吉_n个+1,吉)=0，然后使用引理3.2吉 ∈ F类(年,x个).

定理3.2.让F以下为：X（X）×X（X）→CB（断路器）(X（X）),克以下为：X（X）→X是这样的：定理3.1的条件（1）-（3）成立。设X是完备的，F和g是连续的且相容的。那么，F和g有一个耦合的重合点.

证明.在定理3.1的证明中，我们得到了Cauchy序列{盖克斯 _n个 }和{吉 _n个 }英寸X（X）.自X（X）已完成，存在x个,年 ∈ X（X）使得盖克斯 _n个 →x个和吉 _n个 →年作为n个→∞。最后，我们必须证明这一点盖克斯 ∈ F类(x个,年)和吉 ∈ F类(年,x个). 自从映射以来F类以下为：X（X）×X（X）→CB（断路器）(X（X）)和克以下为：X（X）→X（X）兼容，我们有

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

因为{x个 _n个 }是中的序列X（X），因此x个=极限_n个→∞盖克斯_n个+1∈林_n个→∞F类(x个 _n个 ,年 _n个 )感到满意。对于所有人n个≥0，我们有

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

将限额视为n个→ ∞, 并利用以下事实克和F类是连续的，我们得到，D类(盖克斯,F类(x个,年))=0，这意味着盖克斯 ∈ F类(x个,年).

同样，由于映射F类和克兼容，我们有

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

因为{年 _n个 }是中的序列X（X），因此年=极限_n个→∞吉_n个+1∈林_n个→∞F类(年 _n个 ,x个 _n个 )感到满意。对于所有人n个≥0，我们有

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

取极限为n→ ∞, 并利用以下事实克和F类是连续的，我们得到D类(吉,F类(年,x个))=0，这意味着吉 ∈ F类(年,x个).

由于交换映射是相容的，我们得到了以下结果：；

定理3.3.让F以下为：X（X）×X（X）→CB（断路器）(X（X）),克以下为：X（X）→X是这样的：定理3.1的条件（1）-（3）成立。设X是完备的，F和g是连续的且可交换的。那么，F和g有一个耦合的重合点.