Density of ℝ \ ℚ in ℝ

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

This shows that irrationals are just as ubiquitous as rationals and prevents “gaps” of irrationality; it is often used in constructing sequences with prescribed properties.

Proof sketch (optional): By density of ℚ , choose $q\in\mathbb{Q}$ with $a<q<b$. Then choose a small rational $r>0$ so that $q+r\sqrt{2}\in(a,b)$ (possible since $r\sqrt{2}$ can be made arbitrarily small). The number $q+r\sqrt{2}$ is irrational.