\begin{definition}[Definition:Bounded Metric Space/Definition 1]
Let $M = \struct {A, d}$ be a metric space.
Let $M' = \struct {B, d_B}$ be a subspace of $M$.
'''$M'$ is bounded (in $M$)''' {{iff}}:
:$\exists a \in A, K \in \R: \forall x \in B: \map {d_B} {x, a} \le K$
That is, there exists an element of $A$ within a finite distance of all elements of $B$.
\end{definition}