$f:[0,\infty)\rightarrow\mathbb{R}$ is differentiable and $\lim_{x\to\infty}f'(x)=0$, prove $\lim_{x \rightarrow \infty}[f(x+1)-f(x)]=0$.
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.