\begin{definition}[Definition:Convergent Sequence/Metric Space/Definition 1]
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
$\sequence {x_k}$ '''converges to the limit $l \in A$''' {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$
\end{definition}