Supremum Metric on Continuous Real Functions is Metric

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

Let $\mathscr C \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$.

Let $d$ be the supremum metric on $\mathscr C \left[{a \,.\,.\, b}\right]$.

Then $d$ is a metric.