Supremum Metric on Bounded Real Sequences is Metric/Proof 1

Theorem
Let $A$ be the set of all bounded real sequences.

Let $d: A \times A \to \R$ be the supremum metric on $A$.

Then $d$ is a metric.

Proof
By definition, a real sequence is a mapping from the natural numbers $\N$ to the real numbers $\R$.

Thus a bounded real sequence is a bounded real-valued function.

The result follows from Supremum Metric on Bounded Real-Valued Functions is Metric.