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Thread: cant understand how they use a theorem question..

  1. #1
    MHF Contributor
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    i marked a place where they use theorem 4

    but theorem 4 talks about that if one sequence is bigger then the other
    then so is their liminf and limsup

    but i cant understand how they use this fact for getting the marked result
    what does theorem 4 has to do with the limsup of the sum of sequences
    ??


    why a_n +b_n =limsup(a_n +b_n)
    ??
    Last edited by mr fantastic; Mar 13th 2009 at 03:34 AM. Reason: Merged posts
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  2. #2
    MHF Contributor matheagle's Avatar
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    It's ok, because the limsup of a constant is that constant.
    They have $\displaystyle a_n+b_n<C$, where $\displaystyle C=a+b+\epsilon$.
    Now take the limit sup of both sides of the equation.
    $\displaystyle \limsup a_n+b_n<\limsup C=C$.

    And I pointed out last night that $\displaystyle \limsup (a_n+b_n)\le \limsup a_n +\limsup b_n$.
    This example shows the inequality...
    $\displaystyle a_n=0,1,0,1,0,1...$ and $\displaystyle b_n=1,0,1,0,1,0,...$
    So $\displaystyle \limsup (a_n+b_n)=\limsup \{1,1,1,1,1,1...\}=1$
    while $\displaystyle \limsup a_n +\limsup b_n=1+1=2$.
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  3. #3
    MHF Contributor
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    ok how do i approach this one using the previos method
    $\displaystyle
    limsup(x_n+y_n)=>liminf x_n+limsup y_n
    $
    ??
    Last edited by transgalactic; Mar 13th 2009 at 12:48 AM.
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