The Denseness of Q

Question

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.

Does this answer your question? Infinite number of rationals between any two reals. — Dietrich Burde, Jan 13 '21 at 14:51
You know that there is a rational $r_{n+1}$ such that $a<r_{n+1}<r_n$. — Brian M. Scott, Jan 13 '21 at 17:50

score 1 · Answer 1 · answered Jan 13 '21 at 14:51

You're basically there: there's more than any finite number of rationals between $a$ and $b$ because 1. given any finite list of them you can generate more and 2. $a$ and $b$ themselves compose such a finite list; therefore, the number of rationals between $a$ and $b$ is infinite.

The Denseness of Q

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