Continuous Function on Closed Real Interval is Uniformly Continuous/Proof 1
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Theorem
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a continuous function.
Then $f$ is uniformly continuous on $\closedint a b$.
Proof
We have that $\R$ is a metric space under the usual (Euclidean) metric.
We also have from the Heine-Borel Theorem that $\closedint a b$ is compact.
So the result Heine-Cantor Theorem applies.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.8$: Compactness and Uniform Continuity