Cauchy sequence

A sequence whose terms become arbitrarily close to each other.

Cauchy sequence

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

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

A Cauchy sequence is one that has “intrinsic convergence” without reference to a candidate limit. Completeness of a metric space is the statement that every Cauchy sequence actually converges in the space.

Examples:

In $\mathbb{R}$ , $x_n=1/n$ is Cauchy (and converges to $0$ ).
In $\mathbb{Q}$ with the usual metric, a rational sequence approximating $\sqrt{2}$ is Cauchy but does not converge in $\mathbb{Q}$ .
In a discrete metric space, a sequence is Cauchy iff it is eventually constant.