Prove the following: $f \in L^2(\mathbb{R})$ and $f'$ bounded $\Rightarrow$ $\lim\_{|x| \to \infty}f(x)=0$.

In general, is it also true that $f \in H^1(\mathbb {R} ^n)$ $\Rightarrow \lim\_{|x| \to \infty} f(x)=0$ (where $H^1$ denotes the Sobolev space $W^{1,2}$)?