Difference between revisions of "Cauchy sequence"
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==Definition== | ==Definition== | ||
| − | 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> if: | + | 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: |
| − | * {{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]}} | + | * {{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> |
In words it is simply: | 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. | * 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. | ||
| + | ==See also== | ||
| + | * [[Completeness]] | ||
| + | |||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Functional Analysis|Metric Space|Real Analysis}} | {{Definition|Functional Analysis|Metric Space|Real Analysis}} | ||
Revision as of 15:26, 24 November 2015
Contents
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.
See also
Notes
- ↑ 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
- ↑ 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
