Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 11F / b / iv / Solution

True. First choose polynomials $Q_n\to f$ uniformly by the Weierstrass approximation theorem. Let $L_1,\dots ,L_m$ be the Lagrange interpolation polynomials for the nodes $x_1,\dots ,x_m$, so $L_j(x_i)={\delta }_{ij}$. SetThen $P_n(x_j)=f(x_j)$ for every $j$, while