# Heine Borel Theorem

1. Jun 29, 2011

### zli034

The implications of Heine Borel Thm are not immediate to me. Any results are derived from this theorem?

2. Jun 29, 2011

### micromass

Hi zli034!

The Heine-Borel theorem is so significant that it's used almost everywhere in real analysis. Let me give you an example.

Let's say that B is a ball in $\mathbb{R}^2$. and let's say we have a function $f:B\rightarrow \mathbb{R}$. How do we know that this function has a maximum value? Well, we know that because of the extreme value theorem. This states

If $f:X\rightarrow \mathbb{R}$ is a continuous function and if $X\subseteq \mathbb{R}^n$ is compact, then f has a maximum and a minimum value.​

But how do we know that B is compact? We can prove it directly by showing that every cover has a finite subcover. This is a bit tedious, so we make use of the Heine-Borel theorem which states that it's enough to show that B is closed and bounded.