When I google "baire category theorem", I get a link to Ben Green's website. And at the end of the paper, he mentioned such a classic problem:
Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function with following property: for all $x\in\mathbb{R}^+$, the sequence $f(x), f(2x), f(3x),\dots$ tends to $0$. Prove that $\lim_{t\to\infty}f(t)=0$.
I find the problem 1.17 on P.27 of the book "Selected problems of real analysis", and on P.169 gives the answer by prove the following lemma:
If $G$ is a unbounded open set of $\mathbb{R}^+$, then for any closed interval $[p,q]\ (0
, there exist a $x_0\in [p,q]$ such that $G$ contains infinitely many points of the form $nx_0\ (n\in\mathbb{N})$.
But, on the above book, it also says:
If $\lim_{n\to\infty}f(nx)$ exists only for points $x$ in a nonempty closed set without isolated points, then $\lim_{x\to\infty}f(x)$ also exist.
I didn't find a proof for this result, I want to know whether for any nonempty closed set with no isolated points, the above lemma is true? and could someone tell me why Ben Green mentioned this problem on his paper (see the hint of his paper)?