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

True. By the Weierstrass approximation theorem, choose polynomials $q_n$ with $q_n\to f^{(m)}$ uniformly. Define $P_n$ by integrating $q_n$ exactly $m$ times and choose the $m$ integration constants so thatThen $P_n^{(m)}=q_n$, and repeated use of the fundamental theorem of calculus givesEvery derivative therefore converges uniformly as required.