(a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

(b) Let $\Omega$ be an unbounded, connected, proper open subset of $\mathbb{C}$. For any given compact set $K \subset \mathbb{C} \backslash \Omega$ and any $\zeta \in \Omega$, show that there exists a sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.

(c) Give an example, with justification, of a connected open subset $\Omega$ of $\mathbb{C}$, a compact subset $K$ of $\mathbb{C} \backslash \Omega$ and a point $\zeta \in \Omega$ such that there is no sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.