Suppose that $f’’(x)$ is bounded above on $(0, +\infty)$ and $\lim_{x\to +\infty}f(x)=0$. Prove that $\lim_{x\to +\infty}f’(x)=0$.

Question

Suppose that $f’’(x)$ is bounded above on $(0, +\infty)$ and $\lim\limits_{x\to +\infty}f(x)=0$. Prove that $\lim\limits_{x\to +\infty}f’(x)=0$.

This is my solution:
Consider $h(x)=e^xf(x)$ $\lim\limits_{x\to +\infty}f(x)=\lim\limits_{x\to +\infty} \frac{h(x)}{e^x}=\lim\limits_{x\to +\infty} \frac{h’(x)}{(e^x)’}$ (using L’Hospital’s rule) $=\lim\limits_{x\to +\infty}[f(x)+f’(x)]$ $\Leftrightarrow 0=0+\lim\limits_{x\to +\infty} f’(x)$ hence $\lim\limits_{x\to +\infty}f’(x)=0$.

Actually, I feel that my solution is not really correct. I even haven’t used the condition “ f’’(x) is bounded above”. Can someone help me review and point me the right direction to handle this problem? Thank in advanced.