How to show that $\limsup_{n\to \infty}\frac{a_n}{n}\leq \limsup_{n\to\infty}(a_{n}-a_{n-1})$ for any sequence $(a_n)$?
If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$