No theorem for limit inferior in textbook

Course: Intro to Real Analysis

I'm trying to prove $\displaystyle \text{lim inf }s_n + \text{lim inf }t_n \leq \text{lim inf }(s_n+t_n)$, where $\displaystyle (s_n)$ and $\displaystyle (t_n)$ 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 $\displaystyle (s_n)$ be a bounded sequence. Then the following properties hold:

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

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

Can I say the following?

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

For every $\displaystyle \epsilon >0$ and for every $\displaystyle i \in \mathbb{N}$ there exists an integer $\displaystyle k>i$ such that $\displaystyle s_k <\text{lim inf }s_n+\epsilon$.

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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.

Attachment 30466

Re: No theorem for limit inferior in textbook