Proposition: If a periodic function hasn’t fundamental period then every open interval $)a, b($ contains a period.

My attemption: Let $G$ be the set of periods of $f$. Because of absence basic period $forall x in G$ there is $yin G$ such that $y<frac{x}{2}$. By fixing $x$ and repeating this procedure we can go further and construct other $periods$ less than $frac{x}{2^n} (nin mathbb{N}$) for arbitary higher $n$.

In my opinion choosing arbitary small period isn’t problem. We can do it also by “Dirichlet’s approximation” and may be there are some other ways. But what interests me that is how put $m$ multiple $(min mathbb{N})$ of this period in given interval $)a, b($. It seems a litle bit construction problem.

I would be grateful for help.