Is the set discrete? 
Is the set $S=\{(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}):m,n,p,q\in \mathbb{Z}$} discrete in $\mathbb{R}^2$?
I'm not  getting how shall I check discrete here?


Hint: Take a particular point $P=\left(m+\frac{1}{2^p}, n+\frac{1}{2^q}\right)$.  We want to show that the point is an isolated point. To do this, show that our set contains no point other than $P$ within distance $\min\left(\frac{1}{2^{p+2}},\frac{1}{2^{q+2}}\right)$ of $P$.
