Exercise 1187Topology › Topology of ℝL1Added 3 Nov 2026

Dense sets

#dense#rationals#irrationals

Statement

Show that $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are both dense in $\mathbb{R}$ .

Hint

Between any two reals there is a rational, and between any two reals there is an irrational.

Show solutionHide solution

$\mathbb{Q}$ dense. For $a<b$ reals, pick $n$ with $1/n<b-a$ ; some $k/n\in]a,b[$ .

$\mathbb{R}\setminus\mathbb{Q}$ dense. Given $a<b$ , take a rational $q\in]a,b[$ ; then $q+\frac{\sqrt2}{n}\in]a,b[$ for large $n$ is irrational.

Conclusion. Both $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$ .