I want to show that if $f:\mathbb R\longrightarrow \mathbb R$ (a derivable function) is bounded and s.t. $$\lim\_{x\to \infty }f'(x)=0,$$
then $f$ has a limit in $+\infty $.

I tried as follow:

If $f$ doesn't reach his supremum (let denote it $\ell\in\mathbb R$), then I can construct a sequence $(x\_n)\_n$ s.t. $$\lim\_{n\to \infty }f(x\_n)=\ell.$$

But I can't do better. Any idea ?