[Tao] Ex 5.4.3,

[Tao] Ex 5.4.5 (you may assume result in 5.4.4)

[Tao] Ex 5.4.7,

[Tao] Ex 5.4.8

Let $A$ be the subset of $\Q$ consisting of rational numbers with denominators of the form $2^m$. Prove that for any $x \in \R$, there is a Cauchy sequence $(a_n)$ in $A$, such that $x = LIM a_n$.