Originally Posted by

**tonio** Oh, I have some good hints for you: $\displaystyle x_{2n}\xrightarrow [n\to \infty] {}1\,,\,\,x_{2n-1}\xrightarrow [n\to \infty] {}0$.

You have to show, I'd say by induction, that the sequences with even indexes is monotone increasing and bounded above, and the seq. with odd indexes is monotone decreasing and bounded below, as for both sequences we get the same relation:

$\displaystyle x_{2n}=1-x_{2n-1}^2=1-(1-x_{2(n-1)}^2)^2=2x_{2(n-1)}+x_{2(n-1)}^4$

Since both sequences have limit just use the above relation to get the corresponding limits.

About the second question: nothing... yet.

Tonio