$\{s_n\}$ is sequence of arithmetic means of $\{a_n\}$ Prove that $\liminf a_n \leq \liminf s_n \leq \limsup s_n \leq \limsup a_n$

Question

I thought that it would be useful to prove all these inequalities separately. The second inequality is obvious. Now we have to understand why the first and the third inequations are true either. The idea was to suppose inversed and come to contradiction. But I haven't managed to finish proof using this approach.

Thank you for help in advance!

Answer 1

If $a_n<q$ for all $n>N$ and $a_n<M$ for all $n$, show that $$s_n<\frac{MN+(n-N)q}{n}=q+\frac{NMq}n$$ for all $n>N$ and conclude.