$f$ is Differentiable on all of ${\Bbb R}$ and $\lim_{\ x\to\infty} f^{'}(x) = 0.$ Show $\lim_{\ x\to\infty} (f(x+1)-f(x))=0.$
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.