\begin{definition}[Definition:Bounded Sequence/Metric Space]
Let $M$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Then $\sequence {x_n}$ is a '''bounded sequence''' {{iff}} $\sequence {x_n}$ is bounded in $M$.
That is:
:$\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$
\end{definition}