A sequence {a_{n}} in a metric space is called a Cauchy sequence iff for any (positive) ε, there exists some N=N_{ε} such that d(a_{n},a_{m}) < ε for all n,m ≥ N.

If the metric space is complete, then the limit of the sequence lim a_{n} exists. Proving a sequence is a Cauchy sequence can be easier than showing its limit directly (because we don't need to produce the actual limit!). This is an advantage of working in a complete setting. See the Baire category theorem for a more stunning advantage...