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.
If you have only $n$ finitely many rational numbers between $x$ and $y$, put them in order, $x, r_1, r_2, \ldots r_n, y$.  Can you now prove that there were more than $n$ rational numbers between $x$ and $y$?  If so, you have contradicted your original assumption that there were only $n$ such numbers, which means no such $n$ can exists and there must be infinitely many rational numbers between $x$ and $y$.
Consider the interval $[x,y]$.  Find two rational numbers $q_{min}$ and $q_{max}$ such that $x &lt; q_{min} &lt; q_{max} &lt; y$.  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.