Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2

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

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

Proof
By definition of an open ball:
 * $\forall n \in \N: \map d {x_n, l} < \epsilon \iff x_n \in \map {B_\epsilon} l$

The result follows.