(a) Suppose that $K \subset \mathbb{C}$ is a non-empty subset of the square $\{x+i y: x, y \in(-1,1)\}$ and $f$ is analytic in the larger square $\{x+i y: x, y \in(-1-\delta, 1+\delta)\}$ for some $\delta>0$. Show that $f$ can be uniformly approximated on $K$ by polynomials.

(b) Let $K$ be a closed non-empty proper subset of $\mathbb{C}$. Let $\Lambda$ be the set of $\lambda \in \mathbb{C} \backslash K$ such that $g_{\lambda}(z)=(z-\lambda)^{-1}$ can be approximated uniformly on $K$ by polynomials and let $\Gamma=\mathbb{C} \backslash(K \cup \Lambda)$. Show that $\Lambda$ and $\Gamma$ are open. Is it always true that $\Lambda$ is non-empty? Is it always true that, if $K$ is bounded, then $\Gamma$ is empty? Give reasons.

[No form of Runge's theorem may be used without proof.]