1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Limit Superior - Equivalent Definitions
Loading...