Question

For any positive real number $x$, prove that there exists an irrational number $y$ such that $0<y<x$.

Answer 1

$X$ can be a rational number or irrational number . So let us consider two cases

1) if $X$ is irrational then $\frac{X}{2}$ is also an irrational number which satisfies the hypothesis.

2) if $X$ is rational then $\frac{X}{2}\times \sqrt{2}$ is also an irrational number which satisfies the hypothesis.