# Math Help - lim sup

1. ## lim sup

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

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

2. ## Re: lim sup

$\limsup_n x_n=\lim_n\sup_{k\geq n}x_k=\lim_n\max (\sup_{2k\geq n}x_{2k},\sup_{2k+1\geq n}x_{2k+1})$. Put the $\lim$ into the $\max$ to get the result.