State a version of Runge's theorem and use it to prove the following theorem:

Let $D=\{z \in \mathbb{C}:|z|<1\}$ and define $f: D \rightarrow \mathbb{C}$ by the condition

$$f\left(r e^{i \theta}\right)=r^{3 / 2} e^{3 i \theta / 2}$$

for all $0 \leqslant r<1$ and all $0 \leqslant \theta<2 \pi$. (We take $r^{1 / 2}$ to be the positive square root.) Then there exists a sequence of analytic functions $f_{n}: D \rightarrow \mathbb{C}$ such that $f_{n}(z) \rightarrow f(z)$ for each $z \in D$ as $n \rightarrow \infty$.