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Math Help - Lim sup problem from Halsey Royden, Real Analysis

  1. #1
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    Lim sup problem from Halsey Royden, Real Analysis

    This is my first post here, looking forward to discussing math stuff with everyone. No one in my family knows anything about Real Analysis, so this will be the first time I ever actually talk to anyone about this. I've been plugging along, and I've proved every theorem up to this one, in chapter two section four on sequences of real numbers.

    Royden defines the limit superior as follows:

    \inf_{n} \sup_{k \geq n} x_k\

    He abbreviates the limit superior as:

    \overline{lim}\ x_n

    He defines the limit inferior as sup inf x, as you would expect. The theorem I am trying to prove is:

    \overline{lim}\ x_n\ + \underline{lim}\ y_n\ \leq\ \overline{lim}\ x_n\ +\ y_n.

    I have already proved the Cauchy Criterion and some other useful stuff about cluster points, so I feel like I have a pretty good grasp on this. If someone could just point me in the right direction, I'd appreciate it! I have worked out a working proof, which requires one lemma I don't know how to prove. Anyway, I feel like there's a simpler way to go about this that I'm missing.

    By the way, I have proved the epsilon-N-k definitions of the limit superior, so feel free to use those in your explanation. Thanks again, in advance.

    --Jake
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  2. #2
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    We exclude the case of (\pm\infty)-(\pm\infty). Let a=\limsup x_n, b=\liminf y_n, then there is a subsequence <x_{n_k}> of <x_n> that converges to a. Since \inf \limits_{m \ge {n_k}}y_m\leq y_{n_k}, we have x_{n_k}+\inf\limits_{m \ge {n_k}}y_m\leq x_{n_k}+y_{n_k}. Both x_{n_k} and \inf\limits_{m \ge {n_k}}y_m has limit as k\to\infty (the former is a subsequence of <x_n>, the latter is a subsequence of <\inf \limits_{m \ge n}y_m> which is monotonically increasing and therefore has \sup\limits_{n\ge 1}\inf \limits_{m \ge n}y_m=\liminf y_n as its limit), so the left side approaches a+b as k\to\infty. Suppose c is a subsequential limit of <x_{n_k}+y_{n_k}> (may be \pm\infty) which, as subsequence of subsequence, is also a subsequence of <x_n+y_n>, so a+b\leq c\leq \limsup (x_n+y_n) which is the supremum of the set of all subsequential limits (including possibly \pm\infty).
    Last edited by zzzhhh; June 5th 2010 at 04:59 PM.
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  3. #3
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    Wonderful proof. Thanks for your help. You opened my mind to some better approaches to sequence proofs. I was hindering myself trying to use the epsilon-N-k definition for lim sup.

    By the way, I now understand why lim sup and inf sup are considered the same thing--this is not ambiguous symbolism. I should have seen that fact--that inf sup is a monotonic increasing sequence which would be the same as lim sup. This has really helped me understand it all better.

    --Jake
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