Density of the rationals

Statement

Let $x$ and $y$ be real numbers with $x < y$. Then there is a rational number $q$ with $x < q < y$.

Remarks

Proof idea: choose $n$ with $1/n < y - x$ by the Archimedean property. Then some fraction $m/n$ with $m \in \mathbb{Z}$ lies between $x$ and $y$. Thus between two real numbers there are infinitely many rational numbers. The irrationals are also dense.

Explanations

L1 human

Between any two different real numbers, however close, there is a fraction. For example, between $\sqrt{2} = 1.41421\dots$ and $1.4143$ lies the fraction $1.41425 = 28285/20000$.