Question 125
Let $\{P_n(x)\}_{n \ge 0}$ be a sequence of polynomials given by $P_0(x) = 0$ and
$$
P_{n+1}(x) = P_n(x) + \frac{x - P_n(x)^2}{2}
$$
for $n \ge 0$. Prove that
$$
|P_m(x) - P_n(x)| \lt \frac{1}{n+1}
$$
for all $m \ge n \ge 0$ and all $0 \le x \le 1$.
$$
P_{n+1}(x) = P_n(x) + \frac{x - P_n(x)^2}{2}
$$
for $n \ge 0$. Prove that
$$
|P_m(x) - P_n(x)| \lt \frac{1}{n+1}
$$
for all $m \ge n \ge 0$ and all $0 \le x \le 1$.