Cauchy sequence

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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.

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