Let $S=\{x\in\Bbb{R}~:~1\le|x|\le100\}$. I have to find all subsets $M$ of $S$ such that for all $x,y\in M$, their product $xy$ is also in $M$.
If the MiniMum magnitude among the elements of $M$ is $a$, then the maximum magnitude among the elements of $M$ is $100/a$. But after this I am getting confused. Please help.
Hint. If $M$ has an element of modulus greater than 1 then it is unbounded.