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Use the Runge exhaustion of a slit disk, with the principal argument $-\pi <\arg z<\pi$: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 $K$. ThereforeThe nested exhaustion makes this one sequence work simultaneously on all compact subsets.