Find limits point of the set $A=\left\{ \frac{m}{2^n}| m\in \mathbb{Z}, n\in \mathbb{N}\right\}$

Question

find the limit points of set

$$A=\left\{ \frac{m}{2^n}| m\in \mathbb{Z}, n\in \mathbb{N}\right\}$$

For given any $\alpha \in \mathbb{Z}$ choosing $m=1+ \alpha\ 2^n$ give me $\alpha $ as a limit point.

But it seems that in fact every rational point is limit point of this set . hence this is dense set.

But please provide some hint for proving that every rational point is limit point of this set.

Thanks in advanced.