SCHWEITZER 2015 Problem 2

Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$ .Let $V$ be the set of points on the plane that have the form $(n,x_n)$ .Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ which lies in parallel position with the axes and $R\cap V= \{p,q\}$ .Prove that the chromatic number of $G$ is finite.