Prove that a group is dense in an interval

Question

I tried many ways to solve it and I think it is supposed to be solved using induction, but I seem to get stuck at the very beginning of the induction. I'd like to hear your suggestions of ways to solve it.

Thank you!

Answer 1

HINT: Suppose that $x,y\in(0,1)$ with $x<y$. Choose any $a\in A$ such that $a>1$.

• Explain why there is a positive integer $n$ such that $\frac{a}n<y-x$.
• Show that for each $k\in\Bbb Z^+$, $\frac{ka}n\in B$.
• Show that there must be a $k\in\Bbb Z^+$ such that $x<\frac{ka}n<y$.