Here is what I want to prove.
If $f:(0, \infty) \rightarrow \mathbb{R}$ is continuous and $\displaystyle \lim_{x\rightarrow \infty}[f(x+1)-f(x)]=\alpha$, show that $$\lim_{x\rightarrow\infty} \frac{f(x)}{x}=\alpha$$
Here's what I've tried. I used the $\epsilon-\delta$ definition of limit to convert the given statement to $|f(x+1)-f(x)| < \epsilon$ for large enough $x$, but I don't have any clue on how to proceed.
How do I manipulate this to get something of the form with $f(x)/x$ ?
Any help is greatly appreciated. Thanks in advance!
Note. I've simplified the problem by setting $g(x) = f(x)-\alpha x$, so it is enough to show the problem only for $\alpha = 0$.
