Recursive square root sequence

Let $a_1 =2\pm \sqrt{2}$ and $a_{n+1} =2\pm \sqrt{a_n}$, and let $A_n$ be the set of all such expressions $a_n$. 

(a) Show that all elements of $A_n$ are real. 

(b) Compute the product $$ \prod_{a\in A_n}a$$

(c) If $A_{24}$ issorted in an ascending order, what position is the the element whose signs are $$--++++++++----++++--++-+ $$