Let $\displaystyle (a_n)_{n\in\mathbb{N}}$ be a non-increasing non-negative sequence.

Prove that $\displaystyle s_n= \sum_{i=1}^{n}{(-1)^{i+1}a_i}$ is bounded and that $\displaystyle \limsup{s_n}-\liminf{s_n}=\lim{a_n}$

I would appreciate any help.