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

(b) Suppose $f$ is analytic on

$$\Omega=\{z \in \mathbf{C}:|z|<1\} \backslash\{z \in \mathbf{C}: \operatorname{Im}(z)=0, \operatorname{Re}(z) \leqslant 0\} .$$

Prove that there exists a sequence of polynomials which converges to $f$ uniformly on compact subsets of $\Omega$.