Max and Min of Function on Closed Real Interval/Proof 2

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Then $f$ reaches a maximum and a minimum on $\left[{a \,.\,.\, b}\right]$.

Proof
This is an instance of the Extreme Value Theorem.

$\left[{a \,.\,.\, b}\right]$ is a compact subset of a metric space from Real Number Line is Metric Space.

$\R$ itself is a normed vector space.

Hence the result.