# Definition:Diameter of Bounded Metric Subspace

## Definition

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

Let $S \subseteq A$ be bounded in $M$.

Then the diameter of $S$ is defined as:

$\operatorname {diam} \left({S}\right) := \sup \left\{{d \left({x, y}\right): x, y \in S}\right\}$

That is, by the definition of the supremum, $\operatorname{diam} \left({S}\right)$ is the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.