zli034
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The implications of Heine Borel Thm are not immediate to me. Any results are derived from this theorem?
Heine Borel Theorem: Implications & Results
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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 [itex]\mathbb{R}^2[/itex]. and let's say we have a function [itex]f:B\rightarrow \mathbb{R}[/itex]. How do we know that this function has a maximum value? Well, we know that because of the extreme value theorem. This states
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.
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 [itex]\mathbb{R}^2[/itex]. and let's say we have a function [itex]f:B\rightarrow \mathbb{R}[/itex]. How do we know that this function has a maximum value? Well, we know that because of the extreme value theorem. This states
If [itex]f:X\rightarrow \mathbb{R}[/itex] is a continuous function and if [itex]X\subseteq \mathbb{R}^n[/itex] 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.