L1 Metric on Closed Real Interval is Metric

Theorem
Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $d: S \times S \to \R$ be the $L^1$ metric on $\left[{a \,.\,.\, b}\right]$:
 * $\displaystyle \forall f, g \in S: d \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$

Then $d$ is a metric.

Proof of $M1$
So axiom $M1$ holds for $d$.

Proof of $M2$
So axiom $M2$ holds for $d$.

Proof of $M3$
So axiom $M3$ holds for $d$.

Proof of $M4$
From Zero Definite Integral of Nowhere Negative Function implies Zero Function we have that:
 * $d \left({f, g}\right) = 0 \implies f = g$

on $\left[{a \,.\,.\, b}\right]$.

So axiom $M4$ holds for $d$.