1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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

    bolzano theorem states that each bounded sequence in Rn has a convergent subsequence.

    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


    User Avatar
    Science Advisor

    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!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Heine-Porel from Bolzano-Weierstrass
  1. Isolate y from e (Replies: 14)