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Math Help - lim inf/sup innequality question...

  1. #1
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    lim inf/sup innequality question...

    why this is true
    x_n and y_n are bounded (as n->infinity)
    lim sup x_n >=lim x_r_n
    lim sup y_n >=lim y_r_n
    lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n=lim(x_r_n+y_r_n)
    on what basis they get to lim(x_r_n+y_r_n) from sum of limits??

    and then
    lim(x_r_n+y_r_n)=limsup(x_n +y_n)
    how they got to this conclusion??
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  2. #2
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    x_r_n is a subsequence of x_n
    y_r_n is a subsequence of y_n

    lim(x_r_n+y_r_n)=limsup(x_n +y_n)
    how they got to this conclusion??
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  3. #3
    MHF Contributor
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    in the solution they say that
    because (x_r_n +y_r_n) is convergent then it equals limsup(x_n +y_n)

    so what that it is a convergent?
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