Show that there is a rational number and an irrational number between any two real numbers.

We know that ℚ is dense in ℝ. This means for any x∈ℝ and for any r>0 , ℚ∩B(x;r)≠ϕ where B(x;r)={y∈ℝ:|x−y|<r}

Let a,b∈ℝ$a,b\in \mathbb{R}$ be such that a<b$a<b$. Then we have ℚ∩B(a+b2;b−a3)≠ϕ by above statement. This simply means there is at least one rational number between a$a$ and b$b$.

Similarly, ℝ−ℚis also dense in ℝ By the similar process as above , one can easily derive that there is at least one irrational number between any two distinct real numbers.