$f:(0,\infty) \to \mathbb R$ is continuous, such that $\forall x \in \mathbb R, x >0, \lim_{n \to \infty}f(nx)=0$. I need to show that $\lim_{x \to \infty}f(x)=0$.
I started with an attempt to come with a proof by contradiction: say $x_n \to \infty$ and $f(x_n)$ does not converge to zero, i.e. $\exists \epsilon > 0$ and $x_{n_k}$ sub-sequence of $x_n$ such that $|f(x_{n_k})| > \epsilon, \forall k \in \mathbb N $. From the continuity follows that for each $x_{n_k}$ there is $\delta_{n_k}>0$ such that
$$|f(y)| > \epsilon, \forall y \in (x_{n_k}-\delta_{n_k},x_{n_k}+\delta_{n_k}) $$
And here I am stuck. I would need a hint how to proceed, not a full answer.
