Question
Let $f$ be a differentiable function on $\mathbb {R}$ such that $$\lim\limits_{x \rightarrow -\infty} f(x) = 1\ \mathrm {and} \lim\limits_{x \rightarrow \infty} f(x) = 1.$$
Prove that there exists $c \in \mathbb {R}$ such that $f'(c) = 0$.
My working
Pictorially, I believe it is trivial to see that this is true, as one can just imagine some function that has a horizontal asymptote of $x = 1$.
Mathematically, in general, I know that Rolle's Theorem says that if a differentiable function, $g$, has two points, say, $a$ and $b$, such that $g(a) = g(b) = c$, where $c$ is a real constant, then $\exists\ d \in \mathbb {R}$ such that $g'(d) = 0$. Thus, my intuition is to apply Rolle's Theorem on $f$ and we are done, but I also note that $-\infty$ and $\infty$ are not actually points that $f$ can "reach", so I am not sure if this approach works.
If my intuition is correct, how should I phrase my proof? If it is incorrect, what should be the correct approach to this question? Any help will be greatly appreciated :)