Proving there are infinitely many rational numbers in $[x,y]$ 
Prove if $x$ and $y$ are real numbers with $x \lt y$, then there are infinitely many rational numbers in the interval $[x,y]$.

What I got so far:
Let $x,y \in \Bbb R$ with $x \lt y$ 
Let $S = [x,y]$
By the density of $\Bbb Q$ in $\Bbb R$, $\exists r \in \Bbb Q$ such that $x \lt r \lt y$ where $r \in S$.

This is where I got stuck. 

Consider the interval $[x,y]$.  Find two rational numbers $q_{min}$ and $q_{max}$ such that $x < q_{min} < q_{max} < y$.  Let $r=q_{min}-q_{max}$.  Now make a new number: $q_1 = q_{min} + r/2$.  Let $r_1=q_{max}-q_1$.  Now make a new number: $q_2=q_1+r_1/2$.  Let $r_2=q_{max}-q_2$.  Follow the pattern.  Each of these numbers is a rational number; each number is unique.  The set constructed is countably infinite in size.
