We are given a set $S$ as a subset of the rational numbers defined by:
- $0 \notin S$
- If $s_1 , s_2 \in S$, then $\frac {s_1}{s_2} \in S$
- There exists a nonzero rational number $q \notin S$ such that every nonzero number in $Q \setminus S$ is of the form $qs$ for some $s \in S$.
Prove that if $x \in S$, then there exist $y$ and $z$ in S such that $x=y+z$.
Taking $s_1=s_2$ tells us that $1 \in S$, and thus $ \frac{1}{s} \in S $, and $s^k \in S$ for $k$ an integer and $s \in S$. Perhaps it can be proved by taking $y=x^a, z=x^b$ and proving integers $a$ and $b$ always exist, but I can't figure it out. Can anyone help me?