Supremum Metric on Bounded Real-Valued Functions is Metric/Proof 2

Proof
We have that the supremum metric on $A \times A$ is defined as:


 * $\displaystyle \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in X} \size {\map f x - \map g x}$

where $f$ and $g$ are bounded real-valued functions.

So:
 * $\exists K, L \in \R: \size {\map f x} \le K, \size {\map g x} \le L$

for all $x \in X$.

First note that we have:

and so the exists.

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

Proof of $M2$
Let $f, g, h \in A$.

Let $c \in X$.

Thus $\map d {f, g} + \map d {g, h}$ is an upper bound for:
 * $S := \set {\size {\map f c - \map h c}: c \in X}$

So:
 * $d \left({f, g}\right) + d \left({g, h}\right) \ge \sup S = d \left({f, h}\right)$

So axiom $M2$ holds for $d$.

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

Proof of $M4$
As $d$ is the supremum of the absolute value of the image of the pointwise sum of $f$ and $g$:
 * $\forall f, g \in A: \map d {f, g} \ge 0$

Suppose $f, g \in A: \map d {f, g} = 0$.

Then:

So axiom $M4$ holds for $d$.