Density of the Irrationals

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

Together with the density of the rationals , this implies that every nontrivial interval contains both rational and irrational points, a key intuition behind approximation and continuity arguments.