Let $f : (0, +\infty) \rightarrow \Bbb R$, $f''$ is uniformly continuous and $f$ converges to $0$ as $x$ approaches $\infty$. Show that $f'$ converges to $0$ as $x$ approaches $\infty$.
To be honest, I do not know how to do this, the only thing I know is that $f$ must also be uniformly continuous. How can I prove this?