Cauchy sequence

A sequence in a metric space whose terms become arbitrarily close to each other.

Cauchy sequence

A Cauchy sequence $(x_n)$ in a metric space $(X,d)$ is a sequence such that for every $\varepsilon>0$ there exists $N$ with

d(x_m,x_n)<\varepsilon \quad\text{for all } m,n\ge N.

Every convergent sequence in a metric space is Cauchy (see convergent implies Cauchy ). The converse holds precisely in a complete metric space .

Examples:

In $\mathbb{R}$ , the sequence $x_n=1/n$ is Cauchy.
In $\mathbb{Q}$ with the usual metric, a sequence of rational approximations to $\sqrt{2}$ is Cauchy in $\mathbb{Q}$ but does not converge in $\mathbb{Q}$ .