Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 12F / i / Solution

Let $E=\Vert f-p{\Vert }_{\infty }$. If $E=0$, minimality is immediate. Otherwise suppose a polynomial $q$ of degree at most $n$ has strictly smaller error. At each alternation point,Thus $q-p$ changes sign between every pair of consecutive $x_j$. By the intermediate value theorem it has at least $n+1$ distinct roots of a polynomial, contradicting its degree bound. ThereforeThis is the sufficiency direction of the Chebyshev alternation theorem.