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Math Help - Bolzano–Weierstrass theorem

  1. #1
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    Bolzano–Weierstrass theorem

    I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.
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  2. #2
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    Quote Originally Posted by MathSucker View Post
    I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.
    Some statements of this theorem vary. But here is one source.
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  3. #3
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    What do you think of this one? I want to be able to write it out in the shortest time possible. Is there anything I can trim out?


    If (x_n) is a sequence in the closed segment [a,b] then there exists a subsequence of (x_n) that converges to a limit in [a,b].

    Let c be an arbitrary point in [a,b], then [a,c] can contain and infinite number of members of (x_n) and it can contain a finite number of members of (x_n).

    Let: D=\{c\in[a,b] \, : \,[a,c] contains a finite number of members of (x_n)\}

    If c is in D, then [a,c] has a finite number of members of (x_n), and any subsest of [a,c] can only contain a finite number of members of (x_n). Hence, for any c \in D, all the points between a and c are also in D.

    Let x=sup(D) and assume x \in (a,b).

    Take an arbitrarily small \varepsilon. [a,x+ \varepsilon] is not in D since x+ \varepsilon is greater than sup(D). [a, x - \varepsilon] is in D since x - \varepsilon is less than sup(D).

    [a, x - \varepsilon] is a subset of [a, x + \varepsilon]. The larger set contains an infinite number of members of (x_n) and the subset contains a finite number of members of (x_n). So, the complement of the subset contains an infinite number of members of (x_n).

    Thus, every x in [a,b], has a neighbourhood that contains an infinite number of members of (x_n).

    Set \varepsilon=1. Take any member of (x_n) in (x-1,x+1) and make this the first member of a sequence.

    Set \varepsilon=1/2. Take any member of (x_n) in (x-1/2,x+1/2) with a higher index than the previous member, and make this the next member of the sequence.

    Set \varepsilon=1/4 and so forth indefinitely.

    This results in a subsequence of (x_n) that converges to x.
    Last edited by MathSucker; April 29th 2011 at 09:46 AM.
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  4. #4
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    The best proof based on what? It is, in fact, possible to take "Bolzano-Weierstrasse" as a defining axiom for the real numbers and then the other basic properties (monotone convergence, upper bound property, Heine-Borel, etc.) can be proved from that. But you can also prove "Bolzano-Weierstrasse" from any of them.
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    MHF Contributor Drexel28's Avatar
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    There are very short proofs, but they may use machinery/terms above the level of your current course. What level is this at?
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  6. #6
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    Undergraduate level, but we're given zero worked solutions. The prof basically just writes out proofs on the board, hands us a list of questions, and then walks out. I was just looking for a proof for the "Bolzano-Weierstrasse" theorem that I could bang out on the exam in as little time possible. I found that one, and then wrote it out in my own words and was wondering if it made sense.
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