# Bolzano-Weierstrass theorem

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 $LaTeX: f$ is continuous on its non-empty feasible region, $LaTeX: F$, and that $LaTeX: F$ is a closed set. Then, $LaTeX: f$ achieves a maximum on $LaTeX: F$ if there exists $LaTeX: x^0$ such that $LaTeX: \{ x \in F : f(x) \ge f(x^0) \}$ is bounded.

In particular, if we have a quadratic objective of the form $LaTeX: f(x) = x^T Q x + c^T x$, it is sufficient for $LaTeX: Q$ 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.