Continuous Function on Closed Real Interval is Uniformly Continuous

Theorem
Let $$\left[{a \,. \, . \, b}\right]$$ be a closed real interval.

Let $$f: \left[{a \,. \, . \, b}\right] \to \R$$ be a continuous function.

Then $$f$$ is uniformly continuous on $$\left[{a \,. \, . \, b}\right]$$.

Proof
We have that $\R$ is a metric space under the (usual) Euclidean metric.

We also have from the Heine-Borel Theorem that $$\left[{a \,. \, . \, b}\right]$$ is compact.

So the result Continuous Mapping from Compact Metric Space is Uniformly Continuous‎ applies.