Question 5
Let the polynomial sequence $\{P_n(x)\}_{n=0}^{\infty}$ be defined by $P_0(x)=0$ and
$$P_{n+1}(x)=P_n(x)+\frac{x-P_n^2(x)}2\qquad(n\ge0).$$
Prove that for every $x\in[0,1]$ and every nonnegative integer $n$,
$$0\le\sqrt x-P_n(x)\le\frac2{n+1}.$$
$$P_{n+1}(x)=P_n(x)+\frac{x-P_n^2(x)}2\qquad(n\ge0).$$
Prove that for every $x\in[0,1]$ and every nonnegative integer $n$,
$$0\le\sqrt x-P_n(x)\le\frac2{n+1}.$$