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Showing a sequence is bounded and convergent to its infimum.

  1. Sep 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that any non-increasing bounded from below sequence is convergent to its
    infimum.


    2. Relevant equations

    Not quite sure... is this a monotonic sequence?

    3. The attempt at a solution

    At this point I'm not even sure about which route to go. I am in need of serious help.

    Thanks.
     
  2. jcsd
  3. Sep 18, 2009 #2

    lanedance

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    Homework Helper

    say your sequence is {xi}

    non-increasing means
    xi - xj <0 for all i>j
    which is monotonic

    bounded from below means
    there exists a lower bound a, such that for all i, xi > a

    do you have a theorem about a greatest lower bound existing for bounded below sequence? Otherwise you may have to show this exists, then use the monotnic behaviour of the function to show it converges to the glb{xi) (=inf{xi})
     
    Last edited: Sep 18, 2009
  4. Sep 18, 2009 #3
    Are you speaking of the infimum thm.? I know of that. But it's showing the convergence to it that is the problem for me...
     
  5. Sep 18, 2009 #4

    lanedance

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    so as the sequence is bounded below, you know it has a glb = inf{xi}

    by definition of infinium

    say X = {xi}
    a = inf{xi}

    then for any xi in X , then there exists xj in X such that
    a < xj < xi

    why? & how does it help...? ;)

    what is your defintion of convergence?
     
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