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Thread: Liminf/Limsup of a Sum of Sequences

  1. #1
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    Liminf/Limsup of a Sum of Sequences

    $\displaystyle (a_n)$ and $\displaystyle (b_n)$ are bounded sequences.

    Prove the following:
    $\displaystyle \liminf a_n + \liminf b_n \leq \liminf (a_n + b_n)$
    $\displaystyle \leq \limsup a_n + \liminf b_n \leq \limsup (a_n + b_n)$

    I understand why the second term is greater or equal to the first term but I am having trouble proving mathematically. Mentally, I recognize that they can be equal only when the "liminf terms" of each sequence "line up" (I know this is terrible math terminology ) with each other so they are equivalent to the liminf of the sum. I see similarities between this and the triangle inequality but I can't find a way to apply it.

    As for the other terms in the inequality, I haven't looked at them much yet as I've been hung up on this one. Any thoughts or help with this would be greatly appreciated.
    Last edited by valerian; Nov 4th 2010 at 09:07 PM. Reason: bad formatting
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by valerian View Post
    $\displaystyle (a_n)$ and $\displaystyle (b_n)$ are bounded sequences.

    Prove the following:
    $\displaystyle \liminf a_n + \liminf b_n \leq \liminf (a_n + b_n)$
    $\displaystyle \leq \limsup a_n + \liminf b_n \leq \limsup (a_n + b_n)$

    I understand why the second term is greater or equal to the first term but I am having trouble proving mathematically. Mentally, I recognize that they can be equal only when the "liminf terms" of each sequence "line up" (I know this is terrible math terminology ) with each other so they are equivalent to the liminf of the sum. I see similarities between this and the triangle inequality but I can't find a way to apply it.

    As for the other terms in the inequality, I haven't looked at them much yet as I've been hung up on this one. Any thoughts or help with this would be greatly appreciated.
    How do you define $\displaystyle \limsup$, etc.? The easiest definition for this one is that it's the supremum of the set of all subsequential limits of the sequence.
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  3. #3
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    Yes, that's the definition I'm working with. Liminf is the infinum of all subsequential limits likewise. I'm having trouble working with the terms that are the liminf or limsup of a sum of the sequences.
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