Does there exist an infinite subset, $S$, of the real numbers such that both of the following are true:
- $\forall a,b\in S, \exists c\in S$ such that $a+b=c$
- $\forall c \in S, \exists a,b \in S$ such that $ab\neq c$
My suspicion is that such a set does not exist, but I'm not entirely sure. I know that the real numbers are closed under both addition and multiplication, but I'm not sure if this will help.