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Upper and lower limit proof (liminf/sup)

  1. Sep 3, 2012 #1

    Jesssa

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    Hey,
    So I have been working on this question for quite a while now and I'm at this point.
    I just wanted to check if everything was okay, I never feel confident with questions like this. Here is the question and my working,

    http://img404.imageshack.us/img404/8436/afafafa.jpg [Broken]

    Is there anything wrong?

    Thanks in advanced.
     
    Last edited by a moderator: May 6, 2017
    Jesssa, Sep 3, 2012
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  3. Sep 3, 2012 #2

    voko

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    Are there are conditions on Xn and Yn? The inequality is not true in general. If we set Xn - Yn = a = const, then Xn + Yn = 2, so lim inf {Xn + Yn} = lim sup {Xn + Yn} = a, but lim inf Xn and lim sup Xy may not exist.
     
    Last edited: Sep 3, 2012
    voko, Sep 3, 2012
  4. Sep 3, 2012 #3

    Jesssa

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    The conditions are that they are both bounded real sequences,

    Hmm
     
    Jesssa, Sep 3, 2012
  5. Sep 3, 2012 #4

    voko

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    OK, bounded makes it correct.

    You have the first part correctly.

    The second, however, is wrong. You show that Ix + Sy <= Sx + Sy. That's true, but that does not mean that Ix + Sx <= Sw.
     
    voko, Sep 3, 2012
  6. Sep 4, 2012 #5

    Jesssa

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    I have found another way, do you think this works?

    http://img211.imageshack.us/img211/1769/89852645.jpg [Broken]

    the last part of the first line sup(inf(x)+y) = inf(x) + sup(y), because inf(x) will just be some number,
     
    Last edited by a moderator: May 6, 2017
    Jesssa, Sep 4, 2012
  7. Sep 4, 2012 #6

    voko

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    This looks good to me.
     
    voko, Sep 4, 2012
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