Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 12F / Solution

The set $K$ is compact, and its complement is connected: two disjoint closed disks do not enclose any bounded complementary component. Since $f$ is holomorphic on the open neighbourhood $\Omega$ of $K$, the polynomial Runge theorem gives, for each $n$, a polynomial $p_n$ such thatHence

Define a function on the disconnected open set $\Omega$ byIt is holomorphic on $\Omega$. Applying the first part produces polynomials $P_n$ with

For an obstruction, takeThese are disjoint nonempty bounded closed sets. If polynomials $Q_n$ converged uniformly to zero on $K_1$, the maximum modulus principle on a bounded domain would giveThey therefore cannot also satisfy $Q_n(0)\to 1$. Thus this pair $K_1,K_2$ has the requested non-approximation property.