We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$.
How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial B_{r_k}}Fd\sigma\to0$?
(where $B_{r_k}$ is the open ball:={$x \in \mathbb{R}^N:|x|<r_k$}; $\partial B_{r_K}$:={$x\in\mathbb{R}^N:|x|=r_k$} and $\int_{\partial B_{r_k}}Fd\sigma$ is the integrale surface).
(A book suggest me to try to prove by contradiction)