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Math Help - cant understand what are they doing in this part of a solution..

  1. #1
    MHF Contributor
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    cant understand what are they doing in this part of a solution..

    x_n and y_n are bounded
    the first part was proved
    we have that
    limsup x_n+limsup y_n>=limsup(x_n+y_n)
    now we need to prove that:
    limsup(x_n+y_n)>=liminf x_n +limsup y_n

    as n->infinity
    the say:
    limsup(-x_n)=-liminf(x_n)

    then they say
    limsup(x_n+y_n)-liminf x_n=limsup(x_n+y_n)+limsup (-x_n)>=limsup y_n

    so by putting liminf x_n on the other side we get
    imsup(x_n+y_n)>=liminf x_n +limsup y_n

    why
    limsup(x_n+y_n)+limsup (-x_n)>=limsup y_n

    it should be equal sign
    why its bigger ??
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  2. #2
    MHF Contributor matheagle's Avatar
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    You agree that \limsup a_n+\limsup b_n\ge \limsup (a_n+b_n), right?

    then let a_n=x_n+y_n and b_n=-x_n.

    Note that a_n+b_n=y_n.
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