I'm stuck with the following problem. Show that the map: $$ r(x)=\inf\limits_{k\in\mathbb{N}}\limsup\limits_{m\to\infty}\frac{1}{k}\sum\limits_{j=0}^{k-1}S^j(x)(m) $$ is subadditive on $\ell_\infty(\mathbb{N})$. Here $$ S:\ell_\infty(\mathbb{N})\to\ell_\infty(\mathbb{N}): (x(1),x(2),x(3),\ldots)\mapsto(0,x(1),x(2),x(3),\ldots) $$
Any help greatly appreciated!