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Math Help - Divergence of limit superior

  1. #1
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    Divergence of limit superior

    \lim\sup x_n=-\infty\implies \lim x_n=-\infty, does the converse hold?

    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
    \lim\sup x_n=-\infty\implies \lim x_n=-\infty, does the converse hold?

    How to prove this?
    You tell me, I've helped you with a bunch. Consider maybe the fact that \lim x_n\leqslant \limsup x_n assuming that \lim x_n converges to some element of \mathbb{R}\cup\{-\infty,\infty}
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  3. #3
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    If you understand the definition then this is basically trivial.

    \limsup x_{n} = \lim_{n\to\infty} \sup \{x_{i}\ : i \geq n\}

    Then use the definition of \lim_{n\to\infty}x_{n} = -\infty
    Last edited by Beaky; March 29th 2011 at 06:27 PM.
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