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A metric space is complete if, in it, any Cauchy sequence is convergent.

Intuitively, a space is complete if there are no “points missing” from it, as far as limits of sequences are concerned. For instance, the set of rational numbers is not complete, because e.g. $\sqrt{2}$ is “missing” from it, even though one can construct a sequence of rational numbers that converges to it, which is necessarily a Cauchy sequence. It is always possible to “fill all the holes”, leading to the completion of a given space.

When working on a complete space, one can determine that a sequence converges by proving that it is a Cauchy sequence, thereby avoiding the need of actually determining its limit. This is very useful in Analysis.