Archimedean property of R

Archimedean property of $\mathbb{R}$: For every $x\in\mathbb{R}$ there exists $n\in\mathbb{N}$ such that $n>x$. Equivalently, for every $\varepsilon>0$ there exists $n\in\mathbb{N}$ such that $1/n<\varepsilon$.

This property links the discrete structure of $\mathbb{N}$ to the continuum $\mathbb{R}$ and is used constantly in $\varepsilon$–$N$ arguments, especially to choose large integers making quantities small.

Proof sketch (optional): If the set $\mathbb{N}$ were bounded above , it would have a supremum $s$. Then $s-1$ would not be an upper bound, so some integer $n$ satisfies $n>s-1$, implying $n+1>s$, contradicting that $s$ is an upper bound.