In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. ${\displaystyle {\sqrt {2}}}$ is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.

Cauchy sequence

A sequence ${\displaystyle x_{1},x_{2},x_{3},\ldots }$ of elements from ${\displaystyle X}$ of a metric space ${\displaystyle (X,d)}$ is called Cauchy if for every positive real number ${\displaystyle r>0}$ there is a positive integer ${\displaystyle N}$ such that for all positive integers ${\displaystyle m,n>N,}$ $${\displaystyle d(x_{m},x_{n})<r.}$$

Complete space

A metric space ${\displaystyle (X,d)}$ is complete if any of the following equivalent conditions are satisfied:

The space ${\displaystyle \mathbb {Q} }$ of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit ${\displaystyle x,}$ then by solving ${\displaystyle x={\frac {x}{2}}+{\frac {1}{x}}}$ necessarily ${\displaystyle x^{2}=2,}$ yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number ${\displaystyle {\sqrt {2}}}$.