Prove that there is no such thing as a smallest positive rea

Prove that there is no such thing as a \"smallest positive real number\". [Note: \"positive\" means strictly greater than zero, i.e. z is positive if z > 0.]

Solution

1) Assume the smallest positive real number exists and let it be a. Mathematical representation of the above is.
for every x>0 in R, a<x.
i.e, if some x <a then x is not a real number.

2) Now consider a/2.Its clearly <a. From this we cannot conclude that a/2 is real number because of the implication of assumption

Therefore, there is no such thing as a \"smallest positive real number\"