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

From ProofWiki
Jump to: navigation, search

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.


Hence the result.

$\blacksquare$