If $f$ is differentiable on $[a,\infty)$ and $\lim\limits_{x\rightarrow\infty}f(x)=f(a)$, then $f'$ vanishes at some point.
Your proof is fine. Here is one which is not using the Darboux theorem.
Note that $f$ is continuous.
If $f$ is constant we are done. Otherwise there is $b>a$ such that $f(b) \neq f(a)$, wlog $f(b) > f(a)$. Since $f$ tends to $ f(a)$ there is some $c>b$ such that $f(a) < f(c) < f(b)$ By the mean value theorem for continuous functions there is $d \in (a,b) $ such that $f(d) = f(c)$. Now the claim follows from Rolles Theorem.