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Sup vs. limsup

  1. Jul 18, 2011 #1
    So if you have a countably infinite set [itex]\{ x_n \}[/itex] and consider also the sequence [itex](x_n)[/itex], what's the relationship between [itex]\sup \{ x_n \}[/itex] and [itex]\limsup x_n[/itex]?
     
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  3. Jul 18, 2011 #2

    disregardthat

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    limsup will equal sup if and only there is a subsequence converging to sup. In general sup is always larger or equal to limsup. Both are only well-defined if the sequence is bounded above.

    limsup is the largest value for which there is a subsequence converging to it. In other words it's the largest limit point of the sequence. I may be mistaken, feel free to correct me if I'm wrong.
     
    Last edited: Jul 18, 2011
  4. Jul 18, 2011 #3
    No, this definitely makes sense. I suppose in the case of the sequence [itex]1, 1/2, 1/3, \ldots[/itex], we have [itex]\sup\limits_n x_n = 1[/itex] (since 1 is certainly the least upper bound), but [itex]\limsup\limits_{n\to \infty} x_n = 0[/itex] (since 0 is the only limit point of this set). Thanks!
     
  5. Jul 18, 2011 #4

    disregardthat

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    That seems about right, and no problem.
     
  6. Jul 19, 2011 #5
    I found it very clarifying to introduce the idea of a superior number & an inferior number,
    as is done in this book. Just have a look at the page above the one that comes up in the link.
     
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