Let $S$ be a set of real numbers satisfying the following conditions:
i. $0$ is in $S$.
ii. Whenever $x$ is in $S$ then $2^x+3^x$ is in S.
iii. Whenever $x^2+x^3$ is in $S$ then $x$ is in $S$.
How can I prove that $S$ contains at least two distinct numbers between $0$ and $1$, i.e., (0, 1)?