prove that there exists no smallest possitive real number dose there exist a smallest positive rational number given a real number x, does there exist a smallest real y>x?

Answer 1

pick any real number x > 0.
x/2 < x
so there is always a smaller positive real

same for rational numbers

pick any y such that x < y
The difference is (y-x)

x < x + (y-x)/2 < y