Supremum Metric on Continuous Real Functions is Metric

Theorem
Let $\closedint a b \subseteq \R$ be a closed real interval.

Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.

Let $d$ be the supremum metric on $\mathscr C \closedint a b$.

Then $d$ is a metric.