The Archimedean Property of the Real Numbers

Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, additive identity $0$ and multiplicative inverses $a^{-1}$ of the underlying field and with the order $\le$ of its ordered field structure; for $s,t\in\mathbb{R}$ write $s<t$ to mean $s\le t$ and $s\ne t$. Let $\mathbb{N}$ be the set of natural numbers and let $\iota:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$.

Then the following hold.

1. (Unboundedness) For every $x\in\mathbb{R}$ there exists $n\in\mathbb{N}$ with $x<\iota(n)$.

2. (Archimedean property) For all $x,\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $n\in\mathbb{N}$ with $x<\iota(n)\,\varepsilon$.

3. (Small reciprocals) For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $n\in\mathbb{N}$ such that $\iota(n)^{-1}$ exists and satisfies $0<\iota(n)^{-1}<\varepsilon$.