Supremum Metric on Bounded Real Functions on Closed Interval is Metric/Proof 1
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Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric.
Hence Supremum Metric on Bounded Real-Valued Functions is Metric can be directly applied.