Definition:Convergent Sequence/Metric Space/Definition 1

Definition
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.

Let $\sequence {x_k}$ be a sequence in $A$.

Then $\sequence {x_k}$ converges to the limit $l \in A$ :
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space