Exercise 493Number theory › IrrationalityHigh schoolAdded 3 Nov 2026

Between any two reals lies a rational

#density#rationals#betweenness

Statement

Show that between any two distinct reals, there is a rational.

Hint

Archimedean property: choose $n$ large with $1/n<b-a$ , then find $k$ with $a<k/n<b$ .

Show solutionHide solution

Archimedean. Let $a<b$ reals. Choose $n\in\mathbb{N}^*$ with $1/n<b-a$ .

Integer choice. Consider the integer $k=\lfloor na\rfloor+1$ . Then $k>na$ , so $k/n>a$ , and $k/n=(k-1)/n+1/n\le a+1/n<b$ .

Conclusion. $a<k/n<b$ .

Corollary. Rationals are dense in $\mathbb{R}$ .