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Limit Superior - Equivalent Definitions

  1. Oct 9, 2008 #1
    Let a_1, a_2, ... be a bounded sequence of real numbers. According to Rosenlicht's "Introduction to Analysis", the limit superior is defined as

    sup {x : a_n > x for infinitely many n}.

    It is very hard to work with this definition. I'm used to the simpler one:

    sup {a : there exists a subsequence of a_1, a_2, ... that converges to a}.

    I'm trying to show that these two are equivalent. Denote by A and B the set in the first and second definition, respectively. For any subsequence that converges to a in B, there is a monotonic subsequence, say b_1, b_2, ... that converges to a. If b_1, b_2, ... is increasing, then every b_i is in A. If it is decreasing, then x = inf {b_1, b_2, ...} is in A. For every x in A, there is a corresponding convergent monotonic subsequence. This is all I can think of. Any tips?
  2. jcsd
  3. Oct 9, 2008 #2


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    Try to work with the following intermediate definition:

    [tex]\limsup a_n = \inf_n \sup_{m \ge n} a_m.[/tex]
  4. Oct 9, 2008 #3
    That definition is just as abstruse as Rosenlicht's, at least to me. I thought about it a bit, but I don't see any connection with the other definitions. Here's another thought I have:

    For any collection A' of terms of a_1, a_2, etc., inf A' is in A. So it seems to me that

    sup A = sup { inf A' : A' is a collection of terms of a_1, a_2, ... }

    This seems backwards from the definition you mentioned.
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