A sequence converges if exists.

We can use the monotone convergence theorem to prove the convergence of a sequence.

In Metric Space

Let be an integer, be a metric space, and let as a sequence of points on . Let be a point on . We say that converges to with respect to metric iff the limit exists and is equal to .

Epsilon-delta Definition

see also: epsilon-delta definition converges to with respect to metric iff for every , there exist an such that for all .