I am answering the question:

Consider $a, b ∈ R$ where $a < b$. Use Denseness of $mathbb Q$ to show there are infinitely many rationals between $a$ and $b$.

I have chosen to answer this thing using induction.

I know $P_1$ is a true assertion since by the denseness of $mathbb Q$ there exists a rational, $r_1$ such that $a<r_1<b$. I can then assume $P_n$ is true and that there are $n$ distinct rationals between $a$ and $b$ of the form

$a<r_n<r_{n-1}<r_{n-2}<cdots<r_2<r_1<b$

This is where I’m stuck but I know I want to use the denseness of $mathbb Q$ again to say since $a<r_n$, I can find a rational $r_{n+1}$. At the same time, I don’t know what it is about $mathbb Q$ that allows me to say it.