Proof of the density of the rationals

Proof

1. Let $x$ and $y$ be real numbers with $x < y$. Then $y - x > 0$.
2. By the Archimedean property, there is a natural number $n \ge 1$ with $1/n < y - x$.
3. Let $k = \floor{n x}$ be the floor of $n x$, and let $m = k + 1$. Then $m$ is an integer.
4. By the characterization of the floor, $k \le n x < k + 1$. That is, $m - 1 \le n x < m$.
5. Divide by $n > 0$, which keeps the order in the ordered field $\R$. This gives $m/n - 1/n \le x < m/n$.
6. So $x < m/n$. Also $m/n \le x + 1/n < x + (y - x) = y$, by steps 2 and 5.
7. Let $q = m/n$. It is a rational number, since $m$ and $n$ are integers and $n \ne 0$. By step 6, $x < q < y$.