Let $A=\{z \in \mathbb{C}: 1 / 2 \leqslant|z| \leqslant 2\}$ and suppose that $f$ is complex analytic on an open subset containing $A$.

(i) Give an example, with justification, to show that there need not exist a sequence of complex polynomials converging to $f$ uniformly on $A$.

(ii) Let $R \subset \mathbb{C}$ be the positive real axis and $B=A \backslash R$. Prove that there exists a sequence of complex polynomials $p_{1}, p_{2}, p_{3}, \ldots$ such that $p_{j} \rightarrow f$ uniformly on each compact subset of $B$.

(iii) Let $p_{1}, p_{2}, p_{3}, \ldots$ be the sequence of polynomials in (ii). If this sequence converges uniformly on $A$, show that $\int_{C} f(z) d z=0$, where $C=\{z \in \mathbb{C}:|z|=1\}$.