L1 Metric is Topologically Equivalent to Supremum Metric on Bounded Continuous Real Functions

Theorem
Let $\FF$ be the set of all real functions which are also bounded on the closed interval $\closedint a b$.

Let $d: \FF \times \FF \to \R$ be the $L^1$ metric on $\closedint a b$:
 * $\ds \forall f, g \in \FF: \map d {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$

Let $d': \FF \times \FF \to \R$ be the supremum metric on $\closedint a b$:
 * $\ds \forall f, g \in \FF: \map {d'} {f, g} := \sup_{x \mathop \in S} \size {\map f x - \map g x}$

Then $d$ and $d'$ are topologically equivalent metrics.

Proof
Let $U$ be an upper bound of $\set {\size {\map f x - \map g x} }$.

Then:
 * $\ds U \ge \sup_{x \mathop \in S} \size {\map f x - \map g x}$

Hence:
 * $\ds \max_{x \mathop \in \closedint a b} \set {\size {\map f x - \map g x} } = \map {d'} {f, g}$

Then: