$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the nearest value of $y^2 - y$? I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question.


$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $,  what is the
    nearest value of $y^2 - y$?
$\textbf {(1) } 1 \qquad \textbf {(2) } \sqrt5 \qquad \textbf {(3) } 4 \qquad \textbf {(4) } 2\sqrt5$


I worked the problem and got $$y^2 - y = 5+ \sqrt{5-\sqrt{5+\sqrt{5-...}}} -\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} \\ = 5+ \sqrt{5-y} - y = 5-y + \sqrt{5-y}$$
What do I do next?

Let's let
$$y=\sqrt{5+\sqrt{5-{\sqrt{5+{\sqrt5-\cdots}}}}}$$
and
$$z=\sqrt{5-\sqrt{5+{\sqrt{5-{\sqrt5+\cdots}}}}}$$
and assume that both limits exist.  Note that
$$y^2=5+z\quad\text{and}\quad z^2=5-y$$ 
Hence
$$\begin{align}
y^2-y&=(5+z)-y\\
&=5-(y-z)\\
&=5-{y^2-z^2\over y+z}\\
&=5-{(5+z)-(5-y)\over y+z}\\
&=5-{z+y\over y+z}\\
&=4
\end{align}$$
Added 5/6/14:  I thought I'd go ahead and add a proof that the two limits actually do exist.  I'm not sure this is the simplest proof -- I'll be happy to see something slicker -- but here goes.
Define the sequence of $y_n$'s by $y_n=\sqrt{5+\sqrt{5-y_{n-1}}}$ with $y_1=\sqrt{5+\sqrt{5}}$, and let $z_n=\sqrt{5-y_n}$.  It's easy to show (by induction) that $2\lt y_n\lt4$ for all $n$, so it suffices to show that the sequence of $y_n$'s has a limit.
I'll do so by showing that it's a Cauchy sequence, and I'll do that by showing that $\sum|y_n-y_{n-1}|$ converges.
We have $y_n^2-5=\sqrt{5-y_{n-1}}$, hence
$$y_n^2-y_{n-1}^2=\sqrt{5-y_{n-1}}-\sqrt{5-y_{n-2}}={y_{n-2}-y_{n-1}\over\sqrt{5-y_{n-1}}+\sqrt{5-y_{n-2}}}$$
hence
$$|y_n-y_{n-1}|={|y_{n-2}-y_{n-1}|\over|y_n+y_{n-1}|\left(\sqrt{5-y_{n-1}}+\sqrt{5-y_{n-2}}\right)}\lt{|y_{n-1}-y_{n-2}|\over|2+2|(1+1)}={1\over8}|y_{n-1}-y_{n-2}|$$
where we've used the inequalities $2\lt y_n\lt 4$ (for all $n$) in the denominator.  It follows that $\sum|y_n-y_{n-1}|$ is bounded by a geometric series with ratio $1/8$, hence converges.
Added 6/29/16:  For a nice example of what can go wrong if you assume a limit exists but don't bother to check if it actually does, see this problem.
