<s>[INST] Find $\lim\limits_{n\to\infty}y_n$ if $y_1=\frac{x}{2},y_n=\frac{x}{2}+\frac{y^2_{n-1}}{2},0\le x \le 1,n=2,3,...$ Is it a good approach to use induction? If $0\le x \le 1$ then $0\le y_1 \le \frac{5}{8}$. Suppose that $$0\le y_n \le \frac{5}{8}$$ and prove $$0\le y_{n+1} \le \frac{5}{8}$$ If $$y_{n+1}=\frac{x}{2}+\frac{y^2_n}{2}$$ what should be the induction step?
 [/INST] Since the only possible limits are $1\pm\sqrt{1-x}$, let $z_n=1-y_n$, find the recursion for $z_n$, and check the consequences of $z_n^2<1-x$
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