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I Heine Borel Theorem Proof

  1. Jul 11, 2016 #1
    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?
     

    Attached Files:

  2. jcsd
  3. Jul 11, 2016 #2

    mathman

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    Without the text, the notation you use needs to be defined.
     
  4. Jul 12, 2016 #3
    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?
     

    Attached Files:

  5. Jul 12, 2016 #4

    mathman

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