Let $\displaystyle (s_n)$ be a sequence of nonnegative numbers and for each n define $\displaystyle \sigma_n = \frac{s_1+s_2+...+s_n}{n}$.
show that $\displaystyle \lim \inf s_n \le \lim \inf \sigma_n \le \lim \sup \sigma_n \le \lim \sup s_n$
let $\displaystyle \lim\inf s_n=s$. this means that $\displaystyle \forall\epsilon>0 \exists N$ s.t. $\displaystyle s_n>s-\epsilon$ for all $\displaystyle n\ge N$. then
$\displaystyle \sigma_K=\frac{s_1+\cdots+s_K}{K}=\frac{s_1+\cdots +s_n}{K}+\frac{s_{n+1}+\cdots+s_K}{K}$
$\displaystyle >\frac{s_1+\cdots+s_n}{K}+\frac{(s-\epsilon)(K-N)}{K}>s-1.5\epsilon$ for all large enough K. we showed that $\displaystyle \forall\epsilon>0 \exists K_0$ s.t. $\displaystyle \sigma_K>s-1.5\epsilon$ for all $\displaystyle K\ge K_0$, which implies $\displaystyle \lim\inf \sigma_n\ge s$
middle inequality is trivial, the right inequlaity is done in exactly the same way