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Lim inf/sup innequality question

  1. Mar 12, 2009 #1
    x_n and y_n are bounded
    [tex]
    lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n

    [/tex]
    this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
    the sum of limits is the limit of sums so we get one limit
    and the of two convergent sequence is one convergent sequence
    [tex]
    lim(x_r_n+y_r_n)=limsup(x_n+y_n)
    [/tex]
    this is true because the sequence is constructed from a convergent to the sup sub sequences
    so they equal the lim sup of x_n+y_n

    now i need to prove that
    [tex]
    limsup(x_n+y_n)=>liminf x_n+limsup y_n
    [/tex]

    i tried:
    [tex]
    limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n
    [/tex]
    lim sup i always bigger then lim inf

    so its true

    did i solved it correctly?
     
    Last edited: Mar 12, 2009
  2. jcsd
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