sequence defined by $u_0=1/2$ and the recurrence relation $u_{n+1}=1-u_n^2$ I want to study the sequence defined by $u_0=1/2$ and the recurrence relation $$u_{n+1}=1-u_n^2\qquad n\ge0.$$ I calculated sufficient terms to understand that this sequence does not converge because its odd and even subsequences converge to different limits.
In particular we have $$\lim_{n\to\infty}a_{2n+1}=0\text{ and}\lim_{n\to\infty}a_{2n+2}=1.$$
Now I can also prove that $u_n\leq1$ because the subsequent terms are related in this way $$f\colon x\mapsto1-x^2.$$
In order to prove that the subsequences converge I need to prove either that they are bounded (in order to use Bolzano-Weierstrass) or that they are bounded and monotonic for $n>N$. Proved that the subsequences converge to two different limits, I will be able to prove that our $u_n$ diverges.
Can you help me? Have you any other idea to study this sequence?
Thank you.

The function $f:[0,1]\longrightarrow[0,1]$ has a unique fixed point but is repellent. And we can't reach it in a finite number of steps because is irrational.
