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?

One natural way to complete your proof is by contradiction.
You noticed that, between any two rational numbers, there is a third.
Now, pick any two rational numbers $x < y$, and assume that there are a finite number of (say, only $n$) rational numbers between them. Call these numbers $a_0 < a_1 < \cdots < a_{n-1}$. Now, look between $x$ and $a_0$. We know there is some rational number, call it $z$ between $x$ and $a_0$, which then must also be between $x$ and $y$. But this cannot be true, because $z$ is less than $a_0$, so cannot equal any of the $a_i$, which were assumed to be every rational between $x$ and $y$.
