Let $\displaystyle (s_{n})$ be a sequence of nonnegative numbers, and for each n define $\displaystyle a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$. Show that $\displaystyle \lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_{n} \le \lim \,sup \,s_{n}$. Also show that if $\displaystyle \lim s_{n}$ exists, then $\displaystyle \lim a_{n}$ exists and $\displaystyle \lim a_{n} = \lim s_{n}$.

For this one I am completely stuck. Any help would be appreciated.