Take a function $f:\mathbb{R}\rightarrow(0,+\infty)$ non-decreasing and such that $\mathrm{lim\;inf}_{n\rightarrow+\infty}(f(n+1)-f(n))>0$ then $\mathrm{lim\;sup}_{x\rightarrow+\infty}\;\frac{f(x)}{x}>0$.
The only idea that I got is to prove by contradiction that there is a convergent subsequence of $(f(n+1)-f(n))$ to a limit $\leq0$. But I have no idea how to use this idea, could you help me please?