Cauchy sequence

A sequence whose terms eventually become arbitrarily close to each other

Cauchy sequence

Let $(X,d)$ be a metric space . A sequence $(x_n)$ in $X$ is a Cauchy sequence if

(\forall \varepsilon>0)(\exists N\in\mathbb{N})(\forall m,n\ge N):\ d(x_m,x_n)<\varepsilon.

This definition depends only on the internal mutual distances of the sequence, not on a candidate limit.

Every convergent sequence is Cauchy, but the converse holds exactly in complete metric spaces .

Examples:

In $\mathbb{R}$ , $x_n=1/n$ is Cauchy (and convergent).
In $\mathbb{Q}$ with the usual metric, a sequence of rationals converging to $\sqrt{2}$ is Cauchy but not convergent in $\mathbb{Q}$ .