Density of the Rationals

Density of $\mathbb{Q}$: If $a<b$ are real numbers, then there exists a rational number $q\in\mathbb{Q}$ such that $a<q<b$. In fact, there are infinitely many such rationals.

This is a fundamental approximation tool: it shows that every interval contains “simple” numbers. Standard proofs combine the Archimedean property with basic order properties from the order axioms .