Supremum Metric on Continuous Real Functions is Metric/Proof 1

Proof
Let $\map {\mathscr B} {\closedint a b, \R}$ be the set of all bounded real functions $f: \closedint a b \to \R$.

From Supremum Metric on Continuous Real Functions is Subspace of Bounded, $\struct {\mathscr C \closedint a b, d_{\mathscr C} }$ is a (metric) subspace of $\struct {\map {\mathscr B} {\closedint a b, \R}, d}$.

The result follows from Subspace of Metric Space is Metric Space.