Heine Borel Theorem: Implications & Results

[zli034]

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

 

[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.