I'm trying to prove there is infinite rational numbers between any two rational number I understand how to explain but can't put it down on paper.
$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.
I let $\displaystyle x = \frac{a}{b}$ and $\displaystyle y = \frac{c}{d}$ so $x < y$. the number in the middle of $x$ and $y$ is definitely a rational number so $\displaystyle x < \frac{x+y}{2} < y$. I know that there is another middle between $x$ and $\displaystyle \frac{x+y}{2}$ and it keeps on going. How would I write it as a proof?

You've got the idea. Let $a_1=\frac{x+y}2,$ and $a_{n+1}=\frac{x+a_n}2.$ A quick induction proof will show that each $a_n$ is rational, that $x<a_1<y$ and that $x<a_{n+1}<a_n$ for all $n.$ Thus, you have explicitly determined infinitely-many rational numbers between $x$ and $y.$
More explicitly, you can show by induction that $$a_n=x+\frac{y-x}{2^n}$$ for all $n,$ if you want a closed (non-recursive) form.
