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Math Help - No theorem for limit inferior in textbook

  1. #1
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    No theorem for limit inferior in textbook

    Course: Intro to Real Analysis

    I'm trying to prove \text{lim inf }s_n + \text{lim inf }t_n \leq \text{lim inf }(s_n+t_n), where (s_n) and (t_n) are bounded sequences. There are only theorems for limit superior in our chapter, so I was wondering if I could somehow manipulate the following theorem:

    Let (s_n) be a bounded sequence. Then the following properties hold:

    (a) For every \epsilon >0 there exists a natural number N such that n \geq N implies that s_n<\text{lim sup }s_n+\epsilon.

    (b) For every \epsilon >0 and for every i \in \mathbb{N} there exists an integer k>i such that s_k >\text{lim sup }s_n-\epsilon.

    Can I say the following?

    For every \epsilon >0 there exists a natural number N such that n \geq N implies that s_n>\text{lim inf }s_n-\epsilon.

    For every \epsilon >0 and for every i \in \mathbb{N} there exists an integer k>i such that s_k <\text{lim inf }s_n+\epsilon.
    Last edited by MadSoulz; March 19th 2014 at 05:14 PM.
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  2. #2
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    Re: No theorem for limit inferior in textbook

    Hi,
    Yes. The attachment shows a proof of your statement; this is an exact analog for the corresponding statement about lim sup's. I think the "trick" about using limits when working with lim sup or lim inf is worth remembering.

    No theorem for limit inferior in textbook-mhfcalc40.png
    Thanks from MadSoulz
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  3. #3
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    Re: No theorem for limit inferior in textbook

    Many thanks johng
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