(a) State Runge's theorem about uniform approximability of analytic functions by complex polynomials.

(b) Let $K$ be a compact subset of the complex plane.

(i) Let $\Sigma$ be an unbounded, connected subset of $\mathbb{C} \backslash K$. Prove that for each $\zeta \in \Sigma$, the function $f(z)=(z-\zeta)^{-1}$ is uniformly approximable on $K$ by a sequence of complex polynomials.

[You may not use Runge's theorem without proof.]

(ii) Let $\Gamma$ be a bounded, connected component of $\mathbb{C} \backslash K$. Prove that there is no point $\zeta \in \Gamma$ such that the function $f(z)=(z-\zeta)^{-1}$ is uniformly approximable on $K$ by a sequence of complex polynomials.