For $u \in \mathrm{H}^s\left(\mathbb{R}^n\right)$ with $s>n / 2$, show that $\lim _{x \rightarrow \infty} u(x)=0$.

Question

I was thinking about using Sobolev imbedding theorem. Let $s>m / 2$. Then $$ H^s\left(\mathbb{R}^m\right) \hookrightarrow C_b\left(\mathbb{R}^m\right) . $$

That is $$ H^s\left(\mathbb{R}^m\right) \subset C_b\left(\mathbb{R}^m\right) $$ and there is a constant $C$ such that $\|u\|_{\infty} \leq C\|u\|_{s, 2}$ for every $u \in H^s\left(\mathbb{R}^m\right)$.

I'm unsure how to use this theorem or whether it's the right path to begin with. Do you have any suggestions on how I can approach this problem?