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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$