Let $\displaystyle (x_n)$ be a bounded sequence. For each $\displaystyle n \in \mathbb{N}$ , let $\displaystyle y_n=x_{2n}$ and $\displaystyle z_n=x_{2n-1}$. Prove that

$\displaystyle \limsup{x_n}=\max({\limsup{y_n},\limsup{z_n}})$.

I tried using the identity $\displaystyle \max(x,y)=\frac{1}{2}{(x+y-|x-y|)}$, but it can't seem to work... can anyone help me out?