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

Write $j^{\underline{m}}=j(j-1)\cdots (j-m+1)$ for the falling factorial. At the sampling points, $f_{nm}(j/n)=n^{-m}{j}^{\underline{m}}$, which vanishes for $j<m$. For $j\ge m$, cancellation of the factorials givesInsert this into the Bernstein polynomial and set $\ell =j-m$:The binomial theorem evaluates the remaining sum. For $m=0$, both sides equal one. This proves the Bernstein falling-factorial identity $B_n(f_{nm},x)=f_{nm}(1)x^m$.