Definition:Bounded Metric Space/Definition 2

Definition
Let $M = \left({X, d}\right)$ be a metric space.

Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$.

Then $M'$ is bounded iff:
 * $\exists K \in \R: \forall x, y \in M': d \left({x, y}\right) \le K$

That is, there exists a finite distance such that all pairs of elements of $Y$ are within that distance.

Also see

 * Equivalence of Definitions of Bounded Metric Space