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Math Help - When limit superior diverges

  1. #1
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    When limit superior diverges

    Prove that \lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n so that x_k>M.

    I think is an easy problem, but I'm confused, the statement establishes that x_k is bounded below, but not above.

    How to prove this?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Prove that \lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n so that x_k>M.

    I think is an easy problem, but I'm confused, the statement establishes that x_k is bounded below, but not above.

    How to prove this?
    Which part are you having trouble with? Try firstly the only if statement. Then, you want to prove that \limsup x_n=\infty but evidently by assumption you have that \displaystyle \sup_{n\geqslant N}x_n\geqslant M for every M\in\mathbb{R}^+ so that \displaystyle \limsup x_n=\lim_{N\to\infty}\sup_{n\geqslant N}x_n\geqslant M from where the conclusion follows since M were arbitrary (intuitively you can take the limit as M\to\infty of both sides of this last expression, but the left side is 'unaffected' by the limit since there is no M)
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