The Maximum-minimum theorem is a result about continous functions that deals with a property of intervals rather than that of the function itself.
Let be continous on
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
By the Gap lemma, , such that
As is bounded, by Bolzano-Weierstrass theorem, has a subsequence that converges to
As is continous at ,
R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, John Wiley & Sons