# Bolzano-Weierstrass theorem

### From Glossary

A fundamental existence theorem guaranteeing a maximum and a minimum.

A continuous function on a non-empty compact set achieves a minimum and a maximum over the set.

A useful corollary to the Bolzano-Weierstrass theorem is:

Suppose is continuous on its non-empty feasible region, , and that is a closed set. Then, achieves a maximum on if there exists such that is bounded.

In particular, if we have a quadratic objective of the form , it is sufficient for to be negative definite. Although often credited to Weierstrass alone, it was proven by Bolzano in 1817, and it was known to Cauchy near the end of the 19th century.