Prove that if $p$ and $q$ are rational number with $p < q$ then there exist $ r in A$ such that $p le r le q $

Hint: First notice that if $q-pgeq 1$ then you can just place an integer in between and integers are in $A.$ Then notice that if $p>1$ you can just substract $lfloor p rfloor$ and so without loss of generality you just need to show this for $p<qin [0,1)$ Now play a little game in which you divide the interval by $2$ and if the two numbers are separated you are done and if they are not then you divide the subinterval in which you are. Use the archimedian property to show that there exists an $n$ such that $frac{1}{2^n}<frac{1}{q-p}$ and so the game has to end at some point.