Complete metric space

A metric space in which every Cauchy sequence converges to a point in the space.

Complete metric space

A metric space $(X,d)$ is complete if every Cauchy sequence in $X$ converges to a limit in $X$ . Formally:

\forall (x_n)\subseteq X,\ \Bigl[(x_n)\ \text{Cauchy}\ \Rightarrow\ \exists x\in X\ \text{with}\ x_n\to x\Bigr].

Completeness is a core structural property in analysis: it is needed for many fixed-point and category arguments and is the metric analogue of completeness of $\mathbb{R}$ as an ordered field.

Examples:

$(\mathbb{R},|x-y|)$ is complete.
$(\mathbb{Q},|x-y|)$ is not complete.
Any closed subset of a complete metric space is complete (with the induced metric).