Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$ Find a limit of sequence:
$$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$
$$a_1=0,a_2=0$$
I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by strong induction).
So please give a idea how to show that $a_n$ is monotonic.

(this proof takes a little longer, but gives a neat bound on the rate of convergence)


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*If you want a universal bound of the form $a_n\le M$, the recursion tells you that you're going to need $M\le M^2+\frac{M}{6}+\frac{1}{9}$, which works out to be equivalent to $(6M-1)(3M-2)\le0$. The lowest bound this allows us to take is $M=\frac{1}{6}$, i.e. that $a_n\le\frac{1}{6}$ for all $n$. Feeding this back into the recursion, we see that this inequality does indeed hold.

*If you're hoping for a limit $L$ to exist, the recursion tells you that you're going to need $L=L^2+\frac{L}{6}+\frac{1}{9}$, and similar reasoning to before gives that the only possible limits are $\frac{1}{6}$ and $\frac{2}{3}$. But having established our earlier bounds on $a_n$, we see that if a limit does exist, it will necessarily be $\frac{1}{6}$.

*With the aim of showing that $a_n\to\frac{1}{6}$, we set $a_n=\frac{1}{6}-b_n$ and work through the algebra to obtain the recurrence $b_{n+2}=\frac{b_n}{6}+\frac{b_{n+1}}{3}-b_{n+1}^2$. Our earlier work gives that $0<b_n\le \frac{1}{6}$ for all $n$, and some examination hints that this $b_n$ decays to 0 geometrically, so we seek a bound of the form $b_n\le Cr^n$. Appealing to a strong-induction-style argument, we would need:
$$b_{n+2}=\frac{b_n}{6}+\frac{b_{n+1}}{3}-b_{n+1}^2\le\frac{Cr^n}{6}+\frac{Cr^{n+1}}{3}=Cr^{n+2}\left(\frac{1}{6r^2}+\frac{1}{3r}\right)$$
So, if we can make $\left(\frac{1}{6r^2}+\frac{1}{3r}\right)\le 1$, we can get this inequality to hold (for a suitable $C$) by strong induction. $r=\frac{1+\sqrt7}{6}$ is the smallest $r$ which works for this, and we note that $r\le1$, guaranteeing convergence.
Thus, we see that $0<b_n<\frac{r^{n-2}}{6}$, and hence $b_n\to 0, a_n\to \frac{1}{6}$.
