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Cant understand what are they doing in this part of a solution

  1. Mar 13, 2009 #1
    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 ??
     
  2. jcsd
  3. Mar 13, 2009 #2
    Because in general, they're not equal. For instance, let x_n = 0 if n even, 1 if n odd. Then lim sup (-x_n) = 0, and lim sup(x_n) = 1. Now let y_n = 0 for all n. Then we have 1 = lim sup (x_n+y_n) + lim sup(-x_n) > lim sup y_n = 0.
     
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