No theorem for limit inferior in textbook

• Mar 19th 2014, 04:58 PM
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 $s_k <\text{lim inf }s_n+\epsilon$.
• Mar 19th 2014, 07:25 PM
johng
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.

Attachment 30466
• Mar 19th 2014, 07:55 PM