Let $f$ be continuous such that $$\lim _{x\to \infty }\left(f\left(x+1\right)-f\left(x\right)\right)=0$$ show that $$\lim _{x\to \infty }\left(\frac{f\left(x\right)}{x}\right)=0$$
My idea was applying the sequential criterion for continuity: that if $f$ is continuous at $c$ if and only if for every sequence $x_n$ that converges to $c$, $f(x)$ converges to $f(c)$. Also, I have the idea that since $$\lim_{x\to \infty }\left(f\left(x\right)\right)=\lim_{x\to\infty}\left(f\left(x+1\right)\right),$$ I can use it somehow in the demonstration, or am I wrong?