Supremum Metric on Bounded Real Sequences is Metric/Proof 1
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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.
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.
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$