Definition:Convergent Sequence/Metric Space/Definition 1

Definition
Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space.

Let $\left \langle {x_k} \right \rangle$ be a sequence in $A$.

Then $\left \langle {x_k} \right \rangle$ converges to the limit $l$ iff:
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies d \left({x_n, l}\right) < \epsilon$

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space