Extreme Value Theorem
A continuous real function on a closed bounded set attains its bounds: some satisfy and , so the suprema and infima of are actually maximum and minimum.
Theorem 2.21 (Extreme Value Theorem)
Let be a closed bounded set. If is continuous, then there exist with
Proof
Let . For every , is not an upper bound for , so there is with
Writing with gives a sequence on with . Since is continuous,
By closedness of , the image is closed as well, so : there is with , that is, attained. The infimum case is proved similarly, giving with .
Related
Stated in
- Theorem 2.21 (Extreme Value Theorem)§2.3 Extreme Value Theorem