let $\displaystyle

\left\{ {p_n } \right\}

$ be a sequence of polynomials of real variable defined by

$\displaystyle

p_{n + 1} = \frac{1}{2}(x + 2p_n (x) - \left( {p_n (x)} \right)^2 )

$

show that for all $\displaystyle

x \in \left[ {0,1} \right]

$

$\displaystyle

0 \le \sqrt x - p_n (x) \le \frac{{2\sqrt x }}{{2 + n\sqrt x }}

$

and then show that $\displaystyle

\left\{ {p_n } \right\}

$ converges uniformly to $\displaystyle

f(x) = \sqrt x

$

i need help, i tried to do the first part by induction but i dont seem to get to something useful