Definition:Bounded Metric Space/Definition 1

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 (in $M$) iff:
 * $\exists a \in X, K \in \R: \forall x \in Y: d \left({x, a}\right) \le K$

That is, there exists an element of $X$ within a finite distance of all elements of $Y$.

Also see

 * Equivalence of Definitions of Bounded Metric Space