Max and Min of Function on Closed Real Interval

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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 1

From Image of Closed Real Interval is Bounded, we have that $f$ is bounded on $\closedint a b$.

Let $d$ be the supremum of $f$ on $\closedint a b$.

Consider a sequence $\sequence {x_n}$ in $\closedint a b$ such that $\size {\map f {x_n} } \to d$ as $n \to \infty$.

From the corollary to Limit of Sequence to Zero Distance Point, this can always be found.

Now $\closedint a b$ is a closed interval

So from Convergent Subsequence in Closed Interval, $\sequence {x_n}$ has a subsequence $\sequence {x_{n_r} }$ which converges to some $\xi \in \closedint a b$.

Because $f$ is continuous on $\closedint a b$, it follows from Limit of Image of Sequence that $\map f {x_{n_r} } \to \map f \xi$ as $r \to \infty$.

So $\map f \xi = d$ and thus the supremum $d$ is indeed a maximum.


A similar argument shows that the infimum is a minimum.

$\blacksquare$


Proof 2

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.

$\blacksquare$


Sources

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