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:


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

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

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.