Having some trouble proving this:
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show: $\lim _{x\to \infty } (f(x+1)-f(x))=0$
Attempt: from $\int \lim _{x\rightarrow \infty } f^\prime(x) \mathrm{d}x=0$ we can say that $f^\prime(x)=const$ would we then have to show that $\lim _{x\to \infty } (f(x+1))$ is also a constant? From there if we show that both are equal then the difference is zero. I am on the right track?
Thank you for the help.