Supremum Metric on Continuous Real Functions is Subspace of Bounded

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

Let $d$ be the supremum metric on $\map {\mathscr B} {\closedint a b, \R}$.

Then $\struct {\mathscr C \closedint a b, d_{\mathscr C} }$ is a subspace of $\struct {\map {\mathscr B} {\closedint a b, \R}, d}$.

Proof
Let $f \in \mathscr C \closedint a b$.

Then by Image of Closed Real Interval is Bounded, $f$ is bounded on $\closedint a b$.

Thus $f \in \map {\mathscr B} {\closedint a b, \R}$ and the result follows.