Proof that $\mathbb{Q}$ is dense in $\mathbb{R}$ I'm looking at a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$, using only the Archimedean Property of $\mathbb{R}$ and basic properties of ordered fields.
One step asserts that for any $n \in \mathbb{N}$, $x \in \mathbb{R}$, there is an integer $m$ such that $m - 1 \leq nx < m$.  Why is this true?  (Ideally, this fact can be shown using only the Archimedean property of $\mathbb{R}$ and basic properties of ordered fields...)

If $\mathbb{Q}$ is not dense in $\mathbb{R}$, then there are two members $x,y\in\mathbb{R}$ such that no member of $\mathbb{Q}$ is between them.  I claim that the distance $\varepsilon=|x-y|$ between $x$ and $y$ is an infinitesimal.  By the Archimedean property, this implies $\varepsilon=0$.
If
$$
\underbrace{\varepsilon+\cdots+\varepsilon}_{n\text{ terms}} > 1
$$
then $\varepsilon>1/n$, so some rational number of the form $k/n$ is between $x$ and $y$.
