The following is exercise 13.2 in Rudin's Real & Complex Analysis, which I'm self-studying.
Let $\Omega = \{z: |z| < 1 \text{ and } |2z - 1| > 1\}$, and suppose $f \in H(\Omega)$. Must there exist a sequence of polynomials which converges to $f$ uniformly in $\Omega$?
I have a solution, but I feel it's simple and the shape of $\Omega$ is suspicious so there might be a trick somewhere.
Assume such a sequence exists. Let $0 < \epsilon < 1$, $f(z) = 1/z$ and $P$ be a polynomial that satisfies: $$|f(z) - P(z)| < \epsilon \ \ \forall z \in \Omega$$ Therefore $$|P(z)| < |f(z)| + \epsilon \ \ \forall z \in \Omega$$ But near the boundary of the unit disc, $|f(z)| < 2$, so $|P(z)| < 3$ near the boundary. By the continuity of $P$, we have $P(z) \le 3$ on the boundary. By the maximum modulus principle, $P(z) < 3$ on the unit disc. But $|f(z)|$ gets arbitrarily large near $0$. Therefore $P$ cannot approximate $f$ on $\Omega$.
What gives me confidence in my argument is that it doesn't work on compact subsets of $\Omega$ (for which the existence of the polynomial sequence is guaranteed by Runge's theorem).
Is my counter-example correct?