Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 2F / iii / Solution

If the polynomials converge uniformly on the full annulus to $g$, their limit is continuous and equals $f$ on the slit annulus by part (ii). That subset is dense in the full annulus, so $g=f$ everywhere. The Cauchy integral theorem and uniform convergence on the unit circle then give