$\displaystyle a_{1}=1 $

$\displaystyle a_{n+1}=\sqrt{1+\sqrt{a_{n}}}$

So, taking the limit of both sides, call the limit A, gives us

A=lim$\displaystyle \sqrt{1+\sqrt{a_{n}}}$=$\displaystyle \sqrt{1+\sqrt{A}}$

Solving for A gives us

$\displaystyle A^2=1+\sqrt{A}$ or

$\displaystyle A^2-\sqrt{A}-1=0$

I'm assuming I did all of the above correctly. My only issue now is... how do I solve for A?