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 mathematicianAugustin Louis Cauchy Is named after.
CauchyThe definition of the column depends on the definition of the distance, so only whenmetric space(metric space).In more generalUniform space(uniform space), more abstract Cauchy can be definedFilterCauchy filter and Cauchy net.[1]
One of the important properties is thatComplete space(complete space), all Cauchy columns havelimitThis 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 hasCompletenessThe 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 ofCauchy, if for any positive real number r>0, there is a positiveintegerN makes all integers, both
Formally, given any metric space (M, d), a sequence
go by the name ofCauchy, if for any positive real number r>0, there is a positiveintegerN 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, definitionseries:Meet:。This sequence tends to, butDoes not belong toTherefore, this sequence is not convergent.[2]
For allpolynomialIt 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 anyIt 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 aComplete space。[2]
Example: real number
Real numbers are complete and standardReal number constructioncontainRational numberCausilet.
Counter example: rational number
Rational numberQ is not complete in the sense of commonly defined distance:
There is a sequence composed of rational numbers, which converges to aIrrational numberSo 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 givenFor, the following functionsThe 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 beBoundedSequence.
IfIs a metric space M to metric space NUniform continuityThe mapping of, andCauchy column in M, thenIt must also be Cauchy Column in N.
IfandIs a Cauchy series composed of rational numbers, real numbers or complex numbers, thenandIt's also Cauchy.
extension
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Topological vector space
In aTopological vector spaceX 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 sequencesatisfy, then this sequence is called a Cauchy column.
If there is a topological vector space X, you can introduce aTranslation invariant measured,Then the Cauchy column defined by the above method is equivalent to the Cauchy column defined by the metric d.
orderRepresents the decreasing G of a list of finite indexesNormal subgroup, then a sequence in group GIt is called Cauchy Column (for H above),if and only ifFor 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.andQuotient groupIt is called the completion of G with respect to H.
It can be proved that this complete isomorphism and sequenceOfReverse limitIsomorphism.
If H is aCotailSequence (that is, any finite normal subgroup contains a), then this completion is related toIn the sense of inverse limit isomorphism, H runs through all finite normal subgroups.