Definition:Bounded Metric Space/Definition 1

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

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

$M'$ is bounded (in $M$) :
 * $\exists a \in A, K \in \R: \forall x \in B: d \left({x, a}\right) \le K$

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

Also see

 * Equivalence of Definitions of Bounded Metric Space