This is an exercise from Conway book that I am stuck at. Here I think $G_1$ is contained in $G$. I am required to use the Runge's approximation theorem to solve it. I first thought of the function $1/z$ defined in the punctured disc but $K$ is a compact subset of $G$ so I think $1/z$ can be approximated on $K$ by functions in $H(G)$. Could anyone please help me how to show that such $f$ exists?
Take $f(z)=\frac1z$, defined on $B(0;1)\setminus\{0\}$.