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Homework Help: Heine-Porel from Bolzano-Weierstrass

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data
    How can you prove Heine-Porel (unit interval is compact) theorem by Bolzano-Weierstrass theorem (there is a limit in a continuous and bounded interval)?

    3. The attempt at a solution
    Compact means that the sequence is complete and totally bounded.

    Unit interval perhaps means a bounded interval of an unit length.

    I do not see how the continuity in Bolzano-Weierstrass theorem is related to
    Heine-Porel theorem.
     
  2. jcsd
  3. Mar 3, 2009 #2

    lanedance

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    Homework Helper

    Hi soopo - disclaimer first as i'm fairly new to this stuff myself

    i think the Heine-Borel theorem actually states that a subsapce of Rn is compact iff it is closed & bounded

    defintion of compact is for every open cover, there exists a finite subcover
    http://en.wikipedia.org/wiki/Open_cover

    bolzano theorem states that each bounded sequence in Rn has a convergent subsequence.
    http://en.wikipedia.org/wiki/Bolzano-Weierstrass_theorem

    so start by assuming you have a closed & bounded set eg. [0,1]. Then try & show its compact... and then the other way.. and how can you relate the Bolzano conditions to the compactness cover defintion & can you find a path between them?

    by the way you would have a better chance of getting answered on the calculus part of this forum... Dick & HallsofIvy would rip through it
     
  4. Mar 3, 2009 #3

    HallsofIvy

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    what "continuity" are you talking about? Bolzano-Weierstrasse says that every bounded sequence of real numbers contains a convergent subsequence. That says nothing about "continuity". I don't understand what you mean by "continuous interval"- it might be a language problem- "connected interval"?

    Essentially, the proof that "Bolzano-Weierstrasse" implies "Heine-Borel" uses the fact that a sequence in a closed and bounded set (or, specifically, [0, 1]) is bounded because the set is bouded, so, by Bolzano-Weierstrasse, contains a convergent subsequence. The fact that the set is closed the implies that convergent subsequence converges to a point in the set and so Heine-Borel.
     
  5. Mar 3, 2009 #4
    The reason why B-W theorem implies Heine-Borel is that B-W tells us that a closed and bounded set (eg [0, 1]) contains a convergent subsequence.

    I have been using a continuous interval as a synonym for a connected interval.
    It seems that my convention is false.

    Thank you for your reply!
     
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