Difference between revisions of "Cauchy sequence"

Latest revision as of 21:10, 20 April 2016

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

$\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]$ ^{[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.

There is an equivalence relation which can be defined on Cauchy sequences - see Equivalence of Cauchy sequences

TODO: Flesh this out

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

@@ Line 1: / Line 1: @@
-==Definition==
+{{Stub page|grade=A}}
-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:
+==[[Cauchy sequence/Definition|Definition]]==
-* {{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]}}
+{{:Cauchy sequence/Definition}}
-In words it is simply:
+==Notes==
-* 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.
+* 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/>
-{{Definition|Functional Analysis|Metric Space|Real Analysis}}
+{{Definition|Functional Analysis|Metric Space|Real Analysis|Topology}}