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I have some questions about upper limit

  1. May 6, 2012 #1
    upper limit is defined as:
    lim sup(Sn) = sup E , where {Sn} is a sequence of real numbers and E is the set of all subsequential limits of {Sn}.

    Then if sup E = +∞, why is there a subsequence of {Sn} whose limit is +∞?
    Also, if sup E= -∞ is there a subsequence of {Sn} whose limit is -∞?
  2. jcsd
  3. May 6, 2012 #2


    Staff: Mentor

    From the definition. Slightly restating what you wrote, E is the set of limits of subsequences of {Sn}. This means that some subseqence has a limit of +∞.
    If sup E = -∞, then every subsequence must have this as a limit. Think about it this way: if the largest value in some set is -∞, there's no way to have any smaller (i.e., more negative) values for limits.
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