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 $\closedint a b$.

Then $f$ reaches a maximum and a minimum on $\closedint a b$.

Proof

This is an instance of the Extreme Value Theorem.

$\closedint a b$ is a compact subset of a metric space from Real Number Line is Metric Space.

$\R$ itself is a normed vector space.

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Hence the result.

$\blacksquare$