\begin{definition}[Definition:Convergent Sequence/Metric Space/Definition 4]
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}}:
:for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $x_k$.
\end{definition}