Let $a_1, a_2, a_3,\dotsc$ be a bounded sequence of real numbers. Define $s_n =\frac{(a_1 + a_2 + ...+ a_n)}{n}$, $n \in \mathbb{N}$ Show that $\liminf a_n \le \liminf s_n$
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This is almost the same question as this one: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$. – Martin Sleziak May 07 '13 at 11:47