Question

for any positive real number x prove that there an irrational number y such that 0<y<x

Answer 1

If x is a rational number then $\frac{x}{2}$
is also a rational number and if we multiply that number with $\sqrt{2}$then that would be an irrational number.

$\frac{x}{2}\sqrt{2}=1.414\frac{x}{2}=0.707x$and that would be less than the number x.