A sequence {an} in a metric space is called a Cauchy sequence iff for any (positive) ε, there exists some N=Nε such that d(an,am) < ε for all n,m ≥ N.
If the metric space is complete, then the limit of the sequence lim an 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...