Definition:Convergent Sequence/Metric Space/Definition 3

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:
 * $\displaystyle \lim_{n \mathop \to \infty} d \left({x_n, l}\right) = 0$

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space