Uniformly Continuous Function is Continuous/Real Function/Proof 1

Theorem
Let $I$ be an interval of $\R$.

Let $f: I \to \R$ be a uniformly continuous real function on $I$.

Then $f$ is continuous on $I$.

Proof
From Real Number Line is Metric Space, $\R$ under the Euclidean metric is a metric space.

The result follows by Uniformly Continuous Function is Continuous: Metric Space.