Proof of Density of irrational numbers

Theorem §

Between any two distinct real numbers, there exists an irrational number.

Proof §

Let and be two distinct real numbers with and denotes the positive square root of . Then, we have . By theorem of density of rational numbers, there exists a rational number such that , which implies that . Note that is an irrational number since is irrational. Thus, we have successfully proved that there exists an irrational number between two distinct real numbers.