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Math Help - Supremum

  1. #1
    mus
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    Supremum

    Hi,

    Could someone assist me in the right direction concerning the following problem:

    Deduce the existence of a supremum from the principle of nested intervals.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    I hope this will help...

    Let (a_n) be bounded sequence. Then S=limsup{a_n} exist and he the greater partial limit of the sequence.

    In other words: limsup{a_n}=max\{L:\text{when L is partial limit}\}



    Proof:

    Due to Bolzano-Weierstrass Theorem we may discuss on the set of partial limits, which is non-empty.

    Given \epsilon>0 there are infinite that S-\epsilon<a_n, but only a finite number of S+\epsilon\leq a_n. Therefor there is infinite a_n in neighborhood \epsilon of S, hence S is partial limit.

    Now, suppose that S<S' other partial limit then we will choose \epsilon >0 so that S<S'-\eppsilon, but we know that there is only finite number of S'-\epsilon<a_n, therefor it impossible that S' is partial limit, thus S is the greatest partial limit.
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  3. #3
    mus
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    Could you kindly elaborate on the proof. Its not quite clear.

    Thanks
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