Cauchy sequence

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.

Relation to convergence

TODO: Flesh this out

1. 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