Difference between revisions of "Cauchy sequence"

Latest revision as of 21:10, 20 April 2016

Definition

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.

Notes

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

Relation to 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

1. 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
2. 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

References

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