\text { We define the following polynomials: } P_{0}=0 \text { and } P_{n+1}(x)=P_{n}(x)+\frac{x^{2}-\left[P_{n}(x)\right]^{2}}{2} \quad \text { for all } n \geq 0 \text { (a) Show that } 0

\leq P_{n}(x) \leq P_{n+1}(x) \leq|x| \text { for all }|x| \leq 1 \text { and all } n \geq 0 (b) Show that \lim _{n \rightarrow \infty} P_{n}(x)=|x| \quad \text { uniformly for }|x| \leq 1