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Math Help - lim inf/sup question...

  1. #1
    MHF Contributor
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    lim inf/sup question...

    i got this example:

    http://img405.imageshack.us/img405/2299/64500275qt5.gif

    i got a sequence of numbers
    i cant understand from where they conclude that the limits of every sub sequence is {2,1,0,-1}

    there are no sub sequences pointed out
    there are no formulas for each sub sequence

    how did they came with
    that the limits of every sub sequence is {2,1,0,-1}
    ??
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    i got this example:

    http://img405.imageshack.us/img405/2299/64500275qt5.gif

    i got a sequence of numbers
    i cant understand from where they conclude that the limits of every sub sequence is {2,1,0,-1}

    there are no sub sequences pointed out
    there are no formulas for each sub sequence

    how did they came with
    that the limits of every sub sequence is {2,1,0,-1}
    ??
    If s_n is a convergent subsequence it means the the terms of this sequence have to get sufficiently close to eachother far enough into the sequence (this is Cauchy's criterion). The terms of the full sequence consist of only -1,0,1,2 it means if s_n is a convergent subsequence then it must be eventually constant. Therefore, the only possible subsequential limits are \{ -1,0,1,2\}. The limsup is the largest of these subsequential limits and the liminf is the smallest of these subsequential limits.
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