# lim sup of two sequences

Let $(a_n)_{n \in\ \mathbb{N}}$ a bounded sequence in $\mathbb{R}$. For $n \in \mathbb{N}$ let

$$v_n=\sup\{a_k; ~k \geq n\},\quad u_n=\inf\{a_k; ~k \geq n\},\quad s_n=\sup\{|a_k-a_l|; ~k,l \geq n\}$$

Show that:

(a) $s_n=v_n-u_n$ for all $n \in\ \mathbb{N}$

(b) $\lim\limits_{n \rightarrow \infty} \sup a_n- \lim\limits_{n \rightarrow \infty} \inf a_n = \inf\{s_n;~ n \in\ \mathbb{N}\}$

Unfortunately, i have no idea how to start...