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.
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 $\left\langlex_{n_r}\right\rangle$ (Error compiling LaTeX. Unknown error_msg) 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.