Understanding the Heine Borel Theorem: An In-Depth Analysis

[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_barSecond, for the case 1, why S_gamma+eps does not have a finite subcovering? which definition the author is referring 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?

 

[mathman]

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

 

[kidsasd987]

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

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?

 

[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.