Supremum Metric on Bounded Real-Valued Functions is Metric/Proof 1

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


 * $\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in X} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$

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

From Real Number Line is Metric Space, the real numbers $\R$ together with the absolute value function form a metric space.

The result follows by Supremum Metric is Metric.