# $U\subset [0,\infty)$ is open and unbounded $\Rightarrow \exists x$ such that $U\cap \{nx;n\in \mathbb N\}$ is infinite.

I want to show that:

Let $U\subset [0,\infty)$ be open and unbounded. Show that there is a number $x\in (0,\infty)$ such that $U\cap \{nx;n\in \mathbb N\}$ is infinite.

Because of $U$ is open, $U$ is a countable union of open intervals. if $U$ contain an interval $(a,\infty)$, we are done. But if all intervals which contain in $U$ was bounded, what we can do? can somebody give me a hint?

• This smells like Baire to me – Hagen von Eitzen Nov 17 '15 at 15:11

which is absurd.