Definition:Convergent Sequence/Metric Space/Definition 2

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:
 * $\forall \epsilon > 0: \exists N \in \R_{>0}: n > N \implies x_n \in B_\epsilon \left({l}\right)$

where $B_\epsilon \left({l}\right)$ is the open $\epsilon$-ball of $l$.

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space