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

Statement (iii) is false. Let $\Omega$ be a thin open annulus about the unit circle, let $f(z)=1/z$, and split the circle into two proper closed arcs $K_1,K_2$ whose union is the whole circle. Each arc has connected complement, so polynomial approximation of the reciprocal on a proper circular arc gives uniform polynomial approximations to $f$ on each $K_j$ separately.

If polynomials $P_n$ converged uniformly to $1/z$ on $K_1\cup K_2$, then uniform convergence on the unit circle and the Cauchy integral theorem would givea contradiction.

Solved by gpt-5.6-sol high.