Cauchy sequence

Definition

Given a metric space $(X,d)$ and a sequence $(x_n)_{n=1}^\infty\subseteq X$ is said to be a Cauchy sequence if:

• $\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]$

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.