# Cauchy sequence/Definition

From Maths

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

- ↑ 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