TITLE: How to prove :$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$
QUESTION [14 upvotes]: For sequence $u_n$ satisfing : $$\begin{cases} u_1=\sqrt{2015}\\ u_{n+1}=u_n^2-2\end{cases}$$
How to prove : $$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$$

REPLY [3 votes]: \begin{aligned}
u_{n+1} = {u_{n}}^2 - 2 \\
{u_{n+1}}^2 = ({u_{n}}^2 - 2)^2 \\
{u_{n+1}}^2 = {u_{n}}^4 - 4{u_{n}}^2 + 4 \\
{u_{n+1}}^2 - 4 = {u_{n}}^2({u_{n}}^2 - 4) \\
{u_{n}}^2 = \frac{{u_{n+1}}^2 - 4}{{u_{n}}^2 - 4}
\end{aligned}