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Analysis subsequential limits

  1. Sep 13, 2010 #1
    1. The problem statement, all variables and given/known data

    I missed the last class and I guess our teacher solved this problem and said it would be on our next quiz and I do not really understand how to do it.

    Prove that

    (i) lim sup sn is an element of SL(sn)
    n to infinity

    (ii) same thing but replace the sup with inf

    2. Relevant equations

    2. Relevant equations

    lim sup:= limit supremum
    lim inf:= limit infimum

    3. The attempt at a solution

    I know that it wants me to prove that the limit supremum of a sn is an element of the set of all limits of all convergent subsequences of the sequence sn. It makes sense that the sup and inf of the sequence would be would be elements of the set of all convergent subsequences because in order to have convergent subsequences the original sequence must be bounded. Thus, bounds would seem to be elements of the convergent, bounded subsequences.
    Last edited: Sep 13, 2010
  2. jcsd
  3. Sep 13, 2010 #2


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    Gold Member

    Why don't you try constructing a subsequence which converges to the limit supremum of a bounded sequence? You might need to consider a couple of cases, but it seems like it should be pretty straight forward.
  4. Sep 20, 2010 #3
    I am having trouble showing that there must be a subsequence that converges to the sup any suggestions?
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