ÖMO 1989 Problem 1

Consider the set $S_n$ of all the $2^n$ numbers of the type $2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $2$ appears $n+1$ times. $(a)$ Show that all members of $S_n$ are real. $(b)$ Find the product $P_n$ of the elements of $S_n$ .