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Math Help - Limits and Limit Superior proof

  1. #1
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    Limits and Limit Superior proof



    By definition,
    a_n->a iff
    for all ε>0, there exists an integer N such that n≥N => |a_n - a|< ε





    [note: also under discussion in Math Links forum]
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post


    By definition,
    a_n->a iff
    for all ε>0, there exists an integer N such that n≥N => |a_n - a|< ε





    [note: also under discussion in Math Links forum]
    What is the definition of supremum?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What is the definition of supremum?
    After some more thought, I proved that 1) is true using proof by contradiction. (I always forgot to use the fact that it's the LEAST upper bound). Then taking limits, I got the result because the inequality holds for ALL epsilon>0, so we can get rid of the epsilons.

    How can I prove the other direction? (I am trying to modify my proof but I got stuck)
    Given ε>0.
    a(n)->a => there exists N1 s.t. if n≥N1, then a(n)>a-ε. <-----this is true for sure
    b=limsup b(n) => there exists N2 s.t. if n≥N2, then b(n)> ???

    Can somebody help me, please?
    Last edited by kingwinner; January 26th 2010 at 12:48 AM.
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