Cauchy sequence

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 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]

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.