Subset of Bounded Subset of Metric Space is Bounded

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $B$ be a bounded subset of $M$.

Let $\map \diam B$ denote the diameter of $B$.

Let $C \subseteq B$ be a subset of $B$.

Then $C$ is a bounded subset of $M$ such that:
 * $\map \diam C \le \map \diam B$