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Cauchy sequence

Mathematical concept

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In mathematics, a Cauchy sequence is a sequence whose elements get closer with the increase of the ordinal number. More precisely, after removing a finite number of elements, the maximum value of the distance between any two points in the remaining elements can not exceed any given positive constant. Corsilet is a mathematician Augustin Louis Cauchy Is named after.

Chinese name: Cauchy sequence
Foreign name: cauchy sequence
Scope of application: metric space

founder: Augustin Louis Cauchy
Field: mathematics
Dependent on: Definition of distance

catalog

▪ Topological vector space
▪ In group

brief introduction

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Cauchy The definition of the column depends on the definition of the distance, so only when metric space (metric space). In more general Uniform space (uniform space), more abstract Cauchy can be defined Filter Cauchy filter and Cauchy net.^[1]

One of the important properties is that Complete space (complete space), all Cauchy columns have limit This allows people to prove the existence of the limit by using Cauchy's rule of discrimination without finding out the limit (if it exists). Coxile has Completeness The process of constructing algebraic structure of is also of great value, such as constructing real numbers.

Complex sequence

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A complex sequence

go by the name of Cauchy , if for any positive real number r>0, there is a positive integer N makes all integers

, both

The vertical line indicates absolute value or model 。

Similarly, we can define Cauchy series of real numbers.

In metric space

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In order to extend the definition of Cauchy series to general metric space , you must absolute value Replace with distance 。

Formally, given any metric space (M, d), a sequence

go by the name of Cauchy , if for any positive real number r>0, there is a positive integer N makes all integers m, n>N, have

Where d (x, y) represents the distance between x and y.

Intuitively, the elements in a sequence are getting closer and closer, which seems to indicate that the sequence must have a limit in this metric space. In fact, in some cases, this conclusion is wrong.

example

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For rational number spaces with absolute values as norms

, definition series ：

Meet:

。 This sequence tends to

, but

Does not belong to

Therefore, this sequence is not convergent.^[2]

For all polynomial It defines that the norm of each polynomial is the maximum absolute value of its coefficient, and the distance between two polynomials is the norm of their difference. Consider polynomial columns:

, meet:

。 In this polynomial column, for any

It tends to zero, so it is a Cauchy column. But this Cauchy series is obviously not convergent, because its element number tends to infinity.

nature

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Completeness

All Cauchy sequences in a metric space X will converge to a point in X, then X is called a Complete space 。^[2]

Example: real number

Real numbers are complete and standard Real number construction contain Rational number Causilet.

Counter example: rational number

Rational number Q is not complete in the sense of commonly defined distance:

There is a sequence composed of rational numbers, which converges to a Irrational number So the sequence is not convergent in the space of rational numbers.

For example:

Sequence defined as follows:

, i.e

。 It can be proved that the sequence converges to an irrational number

。

For each given

For, the following functions

The value of can be expressed as the limit of a rational number sequence, but when x is a rational number, this value is an irrational number.

Other properties

Any convergent sequence must be Cauchy sequence, and any Cauchy sequence must be Bounded Sequence.

Is a metric space M to metric space N Uniform continuity The mapping of, and

Cauchy column in M, then

It must also be Cauchy Column in N.

and

Is a Cauchy series composed of rational numbers, real numbers or complex numbers, then

and

It's also Cauchy.

extension

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Topological vector space

In a Topological vector space X can also define a Cauchy column: select a 0 local base B in X, if there is a positive integer N for any element V in B, so that for any m, n>N, the sequence

satisfy

, then this sequence is called a Cauchy column.

If there is a topological vector space X, you can introduce a Translation invariant measure d， Then the Cauchy column defined by the above method is equivalent to the Cauchy column defined by the metric d.

In group

In a group Cauchy column can also be defined:

order

Represents the decreasing G of a list of finite indexes Normal subgroup , then a sequence in group G

It is called Cauchy Column (for H above), if and only if For any r, there is a positive integer N so that for any m, n>N, there is

。

If C is used to represent the set of all Cauchy columns so defined, then C forms a new group in the sense of sequence point multiplication. and

, that is, all empty sequences (for any r, there is N so that for any n>N, there is

）Form the normal subgroup of C. and Quotient group It is called the completion of G with respect to H.

It can be proved that this complete isomorphism and sequence

Of Reverse limit Isomorphism.

If H is a Cotail Sequence (that is, any finite normal subgroup contains a

）, then this completion is related to

In the sense of inverse limit isomorphism, H runs through all finite normal subgroups.

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