Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 3 / c / Solution

Choose a best uniform approximation $p_n\in {\mathcal{P}}_n$. It exists because this vector subspace is a finite-dimensional vector space and a minimizing sequence $q_j$ is bounded: $\Vert q_j{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }+\Vert f-q_j{\Vert }_{\infty }$. Finite-dimensional equivalence of norms then supplies a convergent subsequence of coefficients. If $E_n(f)>0$, the Chebyshev alternation theorem gives $n+2$ alternating error extrema. The intermediate value theorem supplies a zero $x_{n,k}$ of $f-p_n$ in each of the $n+1$ intervening open intervals. These interpolation nodes are distinct, and uniqueness of polynomial interpolation givesThe convergence follows from the Weierstrass approximation theorem. If $E_n(f)=0$, choose any $n+1$ distinct nodes: the polynomial interpolant is still $f=p_n$. This constructs interpolation nodes from best uniform approximation; the nodes depend on $f$, and no universal node system for all continuous functions is asserted.