Archimedean Property

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

This property is a basic consequence of the interaction between the order axioms and the completeness axiom . It is used repeatedly in approximation arguments, for example in the density of the rationals .