Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 62 / 1 / c

Fix $m$ and take $n\ge m$. On $[0,1]$, every factor $x-r/n$ and every factor $x$ has absolute value at most one. Replacing the factors successively and applying the triangle inequality givesUsing the supremum norm contraction in (a) and the Bernstein falling-factorial identity in (b),For $m=0,1$, the monomials are reproduced exactly. For every fixed $m$, $B_n(x^m)\to x^m$ uniformly on $[0,1]$.