Every $f \in H(\Omega)$ can be approximated by polynomials, uniformly on compact subsets of $\Omega$

Question

For a plane region $\Omega$, $H(\Omega)$ denotes the set of holomorphic functions.

$(a)$ To every $f \in H(\Omega)$ corresponds an $F \in H(\Omega)$ such that $F' = f$.

$(b)$ Every $f \in H(\Omega)$ can be approximated by polynomials, uniformly on compact subsets of $\Omega$.

To show $(a) \iff (b).$

I have shown one direction $(b) \implies (a)$, but stuck with the other direction $(a) \implies (b).$

Answer 1

See the section on 'Simply Connected Regions' in Rudin's Real and Complex Analysis. Your a) is g) there and your b) is e). The proof is too lengthy to be reproduced here but if you have difficulty in understanding the proof in Rudin you can post a new question on that.