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Math Help - cant understand how they use a theorem question..

  1. #1
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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; March 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 a_n+b_n<C, where C=a+b+\epsilon.
    Now take the limit sup of both sides of the equation.
    \limsup a_n+b_n<\limsup C=C.

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