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. 

Okay, so I'll give it another shot given the feedback.
Proof:
Let $x,y \in \Bbb R$ with $x \lt y$ and $S = [x,y]$
Suppose there are only $n$ rational numbers between $x$ and $y$ such that:$$x \lt r_1 \lt \cdot \cdot \cdot \lt r_n \lt y$$
But since $\Bbb Q$ is dense in $\Bbb R$, there exists $r_{n+1} \in \Bbb Q$ such that: $$ x \lt r_{n+1} \lt r_1  \lt \cdot \cdot \cdot \lt r_n \lt y$$
which contradicts our assumption that there are only $n$ rational numbers in $[x,y]$.
Therefore, there must be infinitely many rational numbers in the interval $[x,y]$.
