Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 1 / c

Fix $m$ and take $n\ge m$. Comparing the two products defining $f_{nm}$ and the monomial $g_m(x)=x^m$, change one factor at a time. Every factor has absolute value at most one on $[0,1]$, so the triangle inequality givesThe contraction proved in (a) and the Bernstein falling-factorial identity from (b) now giveThis proves uniform convergence for each fixed monomial; linearity then gives it for every fixed polynomial. The cases $m=0,1$ have zero error for every $n$.