Let ($\displaystyle s_n$) be a bounded sequence and let $\displaystyle L =lim sups_n$. Then for every $\displaystyle \epsilon> 0$ , we have $\displaystyle L-\epsilon$ $\displaystyle <s_n$$\displaystyle \le$ $\displaystyle L+\epsilon$ for infinitely many $\displaystyle n$

What would be the lim inf version of this theorem and how would you go about proving it?

Much Thanks!