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?
2026-04-29 23:07:33.1777504053
Find $\lim\limits_{n\to\infty}y_n$ if $y_1=\frac{x}{2},y_n=\frac{x}{2}+\frac{y^2_{n-1}}{2},0\le x \le 1,n=2,3,...$
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Hint: We know that $y_n$ is convergent. Suppose that $y_n\rightarrow y$ then
$$y=x/2+y^2/2\rightarrow y=1-\sqrt{1-x}$$