The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness:

¶ If $S$ is a nonempty subset of $\mathbb{Z}$ and there is $b\in\mathbb{R}$ with $s\le b$ for every $s\in S$, then $S$ has a greatest element.

¶ For every $y\in\mathbb{R}$ and positive $\varepsilon\in\mathbb{R}$ there is $n\in\mathbb{N}$ with $y<n\varepsilon$.

¶ For every positive $\varepsilon\in\mathbb{R}$ there is $n\in\mathbb{N}$ with $0<1/n<\varepsilon$.

Let now $x\in\mathbb{R}$.

¶ There is exactly one $n\in\mathbb{Z}$ with $n\le x<n+1$.

¶ If $n\in\mathbb{Z}$ and $n\le x<n+1$, then $x-1<n$.

¶ If $n\in\mathbb{Z}$, $n\le x<n+1$ and $x\ge0$, then $n\ge0$.

¶ There is $N\in\mathbb{N}$ with $x<n$ for every $n\in\mathbb{N}$ with $n\ge N$.