Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 71 / 1 / a / Solution

Write the Bernstein polynomial in its partition-normalized Bernstein basis:For $0\le x\le 1$, every weight is nonnegative, and the binomial theorem gives their sum as $(x+1-x{)}^n=1$. ConsequentlyTaking the supremum norm proves $\Vert B_n(f){\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }$. The construction yields degree at most $n$, possibly lower for special functions.