VMO 1989 Problem 5

Define the sequence of polynomials $p_0$(x), $p_1$(x), $p_2$(x), ... by $p_0$(x) = 0, $p_{n+1}$(x) = $p_n$(x) + (x - $p_n$(x)$^2$)/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - $p_n$(x) ≤ 2/(n+1).