# Maximum-minimum theorem

The **Maximum-minimum theorem** is a result about continous functions that deals with a property of intervals rather than that of the function itself.

## Contents

## Statement

Let

Let be continous on

Then, has an absolute maximum and an absolute minimum on

## Proof

We will first show that is bounded on ...(1)

Assume if possible such that

As is bounded, is bounded.

By the Bolzano-Weierstrass theorem, there exists a sunsequence of which converges to .

As is closed, . Hence, is continous at , and by the sequential criterion for limits is convergent, contradicting the assumption.

Similarly we can show that is bounded below

Now, Let

By the Gap lemma, , such that

As is bounded, by Bolzano-Weierstrass theorem, has a subsequence that converges to

As is continous at ,

i.e.

## References

R.G. Bartle, D.R. Sherbert, *Introduction to Real Analysis*, John Wiley & Sons