calculus - Prove that $lim_{xrightarrow +infty}left(g(x+sqrt{x})-g(x)right)=0$

Let $g:(1,+\infty)\rightarrow \mathbb{R}$, differentiable, such that $|g'(x)|\leq \frac{1}{x}$ for all $x>1$. Show that

$$\lim_{x\rightarrow +\infty}\left(g(x+\sqrt{x})-g(x)\right)=0.$$

Seems like the mean value theorem is useful. However, not sure how to prove the fact.

Remark:

The hints says that only proving

$$\lim_{x\rightarrow +\infty}|g(x+\sqrt{x})-g(x)|=0.$$
won't suffice.