$\displaystyle\lim_{x\to\infty}f(x)=0$ and $f'(x)\geq0$, then $\displaystyle\lim_{x\to\infty}f'(x)=0$？

Is it correct to say that if a function $f$ of class $C^1$$\displaystyle\lim_{x\to\infty}f(x)=0$ and $f'(x)\geq0$,then $\displaystyle\lim_{x\to\infty}f'(x)=0$?

I attempted to demonstrate this problem using proof by contradiction. Specifically, assuming that $f'(x)$ does not converge to $0$.It implies$\exists \varepsilon>0,\forall M>0,\exists x(x\geq M \ \text{and} \ f'(x)\geq \varepsilon)$.$f$ is class $C^1$, then

• a sequence of disjoint closed intervals $[a_1,b_1],[a_2,b_2],\cdots$ where $f'(x)\geq \dfrac{\varepsilon}{2}$ exists.
• $f(x)=\displaystyle\int_a^xf'(t)dt+f(a)$

I tried using these two, attempting to demonstrate that $f$ has a positive lower bound, but I couldn't.