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Limit superior: proof of properties.

  1. Feb 25, 2009 #1
    I am only stuck on one part of one fairly long proof.

    1. The problem statement, all variables and given/known data

    Let {a_n} be a sequence in R. Then the following are equivalent:
    (a) limsup a_n = a.
    (b) For every b > a, a_n < b for all but finitely many n and for every c < a, a_n > c for infinitely many n.

    assume that (b) holds. Then for every b > a, there exists m such that an < b for all n ≥ m. Hence sup_n≥m a_n ≥ b. This implies that limsup a_n ≥ b for every b > a so that limsup a_n ≥ a.

    3. The attempt at a solution

    How does the limsup a_n ≥ b inequality hold? As far as I understand limsup it is a the limit of a monotonically decreasing sequence, sup_n≥m a_n, which cannot have a limit that is larger than the largest member of the sequence. But we know that b is larger. So shouldn't the limsup a_n ≥ b be a strict inequality because of this?
  2. jcsd
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