# I Heine Borel Theorem Proof

1. Jul 11, 2016

### kidsasd987

Hello, I have a question about Heine Borel Theorem.

First, I am not sure why we have to show
"gamma=Beta"
gamma is the supremum of F(which is equivalent to H_squiggly_bar in the text ), and it has to be greater than beta. Otherwise, S contains H_squiggly_bar

Second, for the case 1, why S_gamma+eps does not have a finite subcovering? which definition the author is reffering to?

I understand sup(F) = gamma, so S_gamma-eps must have a finite subcovering because by definition H_squiggly_bar is a set of finite subcovering. But isn't there a possibility that S_gamma+eps also has a finite subcovering?

That consists of H_squiggly bar + some finite set that belongs to H but not contained within H_squiglly bar?

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• ###### Heine_Borel_Theorem.png
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2. Jul 11, 2016

### mathman

Without the text, the notation you use needs to be defined.

3. Jul 12, 2016

### kidsasd987

I am sorry. This is the updated version. Do you think the proof is right? and what do you think of the question at the end?

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• ###### Heine_Borel_2.png
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4. Jul 12, 2016

### mathman

I am finding these attachments hard to read (type size). In the latest attachment, much of the analysis makes use of $H_i$, which is not defined.