State Runge's theorem on the approximation of analytic functions by polynomials.

Let $\Omega=\{z \in \mathbb{C}, \operatorname{Re} z>0, \operatorname{Im} z>0\}$. Establish whether the following statements are true or false by giving a proof or a counterexample in each case.

(i) If $f: \Omega \rightarrow \mathbb{C}$ is the uniform limit of a sequence of polynomials $P_{n}$, then $f$ is a polynomial.

(ii) If $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there exists a sequence of polynomials $P_{n}$ such that for each integer $r \geqslant 0$ and each $z \in \Omega$ we have $P_{n}^{(r)}(z) \rightarrow f^{(r)}(z)$.