Definition:Convergent Sequence/Metric Space/Definition 2

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 $l \in A$ :
 * $\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$

where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.

Also see

 * Equivalence of Definitions of Convergent Sequence in Metric Space