= Solution
Use the <Runge exhaustion of a slit disk>, with the principal argument $-\pi<\arg z<\pi$:
$$
K_m=\{re^{i\theta}:1/m\leq r\leq1-1/m,\quad |\theta|\leq\pi-1/m\},\qquad m\geq3.
$$
These are nested compact subsets of $\Omega$. Their complements are connected: the inner disk and exterior region join through the omitted sector around the negative real radius. Every compact subset of $\Omega$ has positive distance from zero, the unit circle and the slit, so it is contained in $K_m$ for all sufficiently large $m$.
Since $f$ is <holomorphic> on $\Omega$, it is <holomorphic> on a neighborhood of each $K_m$. The <polynomial Runge theorem> gives $P_m$ with $\sup_{K_m}|P_m-f|<1/m$. Given any compact $K\subset\Omega$, eventually $K\subseteq K_m$, so the same estimate holds on $K$. Therefore
$$
\boxed{P_m\longrightarrow f\text{ uniformly on every compact subset of }\Omega.}
$$
The nested exhaustion makes this one sequence work simultaneously on all compact subsets.
Back to article page
