Density of ℚ in ℝ

Density of $\mathbb{Q}$ in $\mathbb{R}$: If $a<b$ are real numbers, then there exists $q\in\mathbb{Q}$ such that $a<q<b.$

This ensures rationals approximate reals arbitrarily well and is foundational for approximation arguments, constructions via sequences, and separating points in analysis.

Proof sketch (optional): Choose $n\in\mathbb{N}$ with $n(b-a)>1$ (Archimedean property ). Then pick an integer $m$ with $na<m<nb$ (using existence of integers between reals). Set $q=m/n$.