Difference between revisions of "Cauchy sequence"
From Maths
(→Definition) |
m (Marked as stub page, added mention of equivalence relation definable on Cauchy sequences) |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | == | + | {{Stub page|grade=A}} |
| − | + | ==[[Cauchy sequence/Definition|Definition]]== | |
| − | * {{ | + | {{:Cauchy sequence/Definition}} |
| − | * | + | ==Notes== |
| − | + | * There is an [[equivalence relation]] which can be defined on ''Cauchy sequences'' - see ''[[Equivalence of Cauchy sequences]]'' | |
| + | ==Relation to [[Convergence (sequence)|convergence]]== | ||
| + | * [[Every convergent sequence is Cauchy]] and | ||
| + | * [[In a complete metric space every Cauchy sequence converges]] | ||
| + | {{Todo|Flesh this out}} | ||
| + | ==See also== | ||
| + | * [[Convergence of a sequence]] | ||
| + | * [[Completeness]] | ||
| + | * [[Equivalence of Cauchy sequences]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{Definition|Functional Analysis|Metric Space|Real Analysis}} | + | {{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
- There is an equivalence relation which can be defined on Cauchy sequences - see Equivalence of Cauchy sequences
Relation to convergence
TODO: Flesh this out
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
References
- ↑ Functional Analysis - George Bachman and Lawrence Narici
- ↑ Analysis - Part 1: Elements - Krzysztof Maurin
