Supremum Metric on Bounded Real Functions on Closed Interval is Metric

Theorem
Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval.

Let $A$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$.

Let $d: A \times A \to \R$ be the supremum metric on $A$.

Then $d$ is a metric.