Editing Cauchy sequence (section)

{{Short description|Sequence of points that get progressively closer to each other}}
{{Use shortened footnotes|date=November 2022}}
{{multiple image
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|direction = vertical
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|image1    = Cauchy sequence illustration.svg
|caption1  = (a) The plot of a Cauchy [[sequence (mathematics)|sequence]] <math>(x_n),</math> shown in blue, as <math>x_n</math> versus <math>n.</math> If the [[Space (mathematics)|space]] containing the sequence is [[Complete metric space|complete]], then the sequence has a [[Limit (mathematics)|limit]].
|image2    = Cauchy sequence illustration2.svg
|caption2  = (b) A sequence that is not Cauchy. The [[Element (mathematics)|elements]] of the sequence do not get arbitrarily close to each other as the sequence progresses.
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In [[mathematics]], a '''Cauchy sequence''' is a [[sequence]] whose [[Element (mathematics)|elements]] become arbitrarily close to each other as the sequence progresses.{{sfn|Lang|1992}} More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after [[Augustin-Louis Cauchy]]; they may occasionally be known as '''fundamental sequences'''.<ref>{{cite book|first=Heinz-Dieter |last=Ebbinghaus|title=Numbers|place=New York|publisher=Springer|date=1991|page=40}}</ref>

It is not sufficient for each term to become arbitrarily close to the {{em|preceding}} term. For instance, in the sequence of square roots of natural numbers:
<math display="block">a_n=\sqrt n,</math>
the consecutive terms become arbitrarily close to each other – their differences
<math display="block">a_{n+1}-a_n = \sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2\sqrt n}</math>
tend to zero as the index {{mvar|n}} grows. However, with growing values of {{mvar|n}}, the terms <math>a_n</math> become arbitrarily large. So, for any index {{mvar|n}} and distance {{mvar|d}}, there exists an index {{mvar|m}} big enough such that <math>a_m - a_n > d.</math>  As a result, no matter how far one goes, the remaining terms of the sequence never get close to {{em|each other}}; hence the sequence is not Cauchy.

The utility of Cauchy sequences lies in the fact that in a [[complete metric space]] (one where all such sequences are known to [[Limit of a sequence|converge to a limit]]), the criterion for [[Convergence (mathematics)|convergence]] depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in [[algorithm]]s, both theoretical and applied,  where an [[Iterative method|iterative process]] can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. 

Generalizations of Cauchy sequences in more abstract [[uniform spaces]] exist in the form of [[Cauchy filter]]s and [[Cauchy net]]s.