Proof
Let be arbitrary. Note that and are real numbers. Then by density of rationals, there exists such that Then note that is irrational. Thus, there exists an irrational number between two real numbers.
Let a,b∈R be arbitrary. Note that a−π and b−π are real numbers. Then by density of rationals, there exists r∈Q such that a−π<r<b−π Then a<r+π<b note that r+π is irrational. Thus, there exists an irrational number between two real numbers. □