Let $\Gamma=\{\varphi^n\mid n\in\mathbb{N}\}$ where $\varphi(x,y)=(\frac{x}{2^n},2^ny)$. I am trying to decide if $\Gamma$ defines a properly discontinuous action on $X=(0,\infty)\times(0,\infty)$. I am stuck trying to find neighbourhoods $U,V$ of $x,y\in X$ that $$\{n\in\mathbb{N}\mid \varphi(U)\cap V\neq \emptyset\}$$ is finite.
I have been trying to draw it by finding a fundamental domain, so I can intuitively see how does this action work. But I can't manage to do it.