Difference between revisions of "Cauchy sequence"

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==Definition==
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{{Stub page|grade=A}}
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Given a [[Metric space|metric space]] {{M|(X,d)}} and a [[Sequence|sequence]] {{M|1=(x_n)_{n=1}^\infty\subseteq X}} is said to be a ''Cauchy sequence''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref><ref name="KMAPI">Krzysztof Maurin - Analysis - Part I: Elements</ref> if:
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==[[Cauchy sequence/Definition|Definition]]==
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* {{M|\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]}}<ref group="Note">Note that in [[Krzysztof Maurin's notation]] this is written as {{MM|1=\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{m,n>\mathbb{N} }d(x_n,x_m)<\epsilon}} - which is rather elegant</ref><ref group="Note">It doesn't matter if we use {{M|n\ge m>N}} or {{M|n,m\ge N}} because if {{M|1=n=m}} then {{M|1=d(x_n,x_m)=0}}, it doesn't matter which way we consider them (as {{M|n>m}} or {{M|m>n}}) for {{M|1=d(x,y)=d(y,x)}} - I use the ordering to give the impression that as {{M|n}} goes out ahead it never ventures far (as in {{M|\epsilon}}-distance}}) from {{M|x_m}}. This has served me well</ref>
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{{:Cauchy sequence/Definition}}
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In words it is simply:
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==Notes==
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* For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.
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* There is an [[equivalence relation]] which can be defined on ''Cauchy sequences'' - see ''[[Equivalence of Cauchy sequences]]''
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==Relation to [[Convergence (sequence)|convergence]]==
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* [[Every convergent sequence is Cauchy]] and
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* [[In a complete metric space every Cauchy sequence converges]]
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{{Todo|Flesh this out}}
 
==See also==
 
==See also==
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* [[Convergence of a sequence]]
 
* [[Completeness]]
 
* [[Completeness]]
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* [[Equivalence of Cauchy sequences]]
 
==Notes==
 
==Notes==
 
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{{Definition|Functional Analysis|Metric Space|Real Analysis}}
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{{Definition|Functional Analysis|Metric Space|Real Analysis|Topology}}

Latest revision as of 21:10, 20 April 2016

Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.

Definition

Given a metric space [ilmath](X,d)[/ilmath] and a sequence [ilmath](x_n)_{n=1}^\infty\subseteq X[/ilmath] is said to be a Cauchy sequence[1][2] if:

  • [ilmath]\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon][/ilmath][Note 1][Note 2]

In words it is simply:

  • For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.

Notes

Relation to convergence

TODO: Flesh this out

See also

Notes

  1. ↑ Note that in Krzysztof Maurin's notation this is written as [math]\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{m,n>\mathbb{N} }d(x_n,x_m)<\epsilon[/math] - which is rather elegant
  2. ↑ It doesn't matter if we use [ilmath]n\ge m>N[/ilmath] or [ilmath]n,m\ge N[/ilmath] because if [ilmath]n=m[/ilmath] then [ilmath]d(x_n,x_m)=0[/ilmath], it doesn't matter which way we consider them (as [ilmath]n>m[/ilmath] or [ilmath]m>n[/ilmath]) for [ilmath]d(x,y)=d(y,x)[/ilmath] - I use the ordering to give the impression that as [ilmath]n[/ilmath] goes out ahead it never ventures far (as in [ilmath]\epsilon[/ilmath]-distance}}) from [ilmath]x_m[/ilmath]. This has served me well

References

  1. ↑ Functional Analysis - George Bachman and Lawrence Narici
  2. ↑ Analysis - Part 1: Elements - Krzysztof Maurin
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