Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 2F / iv / Solution

Statement (iv) is true. Apply the Weierstrass approximation theorem separately to the real and imaginary parts of the continuous function $f$ on the compact interval $[0,1]$. Combining the two real polynomial sequences gives complex polynomials converging uniformly to $f$ on $I$.