Runge's theorem says that an analytic function $f$ in an open set containing compact set $K$ can be approximated by a rational function with poles in $E$,where $E$ is a subset of $\mathbb{C}_\infty-K$,and more importantly,$E$ meets every component of $\mathbb{C}_\infty-K$.

Now the problem is for any compact set $K$ contained on an open set $G_1\subset G$,can functions in $H(G_1)$ be approximated on $K$ by functions in $H(G)$?If the choice of $K,G,G_1$ is arbitrary,of course it's false,such as an annulus.We are now trying to establish a criteria for a fixed $K$ and $G$ such that this can be done for any $G_1$.

Thus we need to prove the equivalence of following three statements:
(a)If $f$ is analytic in a neighborhood of $K$ and $\epsilon >0$ then there is a $g$ in $H(G)$ with $|f(z)-g(z)|<\epsilon$ for all $z$ in $K$;
(b)If $D$ is a bounded component of $G-K$ then $D^{-}\cap G\neq \emptyset$;
(c)If $z$ is any point in $G-K$ then there is a function $f$ in $H(G)$ with $|f(z)|>sup\{|f(\omega)|:\omega \in K\}$;

$(b)\Rightarrow (a)$ is directly from Runge's Theorem when noticing that $(b)$ implies every component of $\mathbb{C}_\infty-K$ contains a component of $\mathbb{C}_\infty-G$ .
I found it difficult to prove $(a)\Rightarrow(c)$,in which $(c)$ asserts that holomorphic convex hull of $K$ in $G$ is $K$ itself.I don't know how to build connection between Runge's theorem and convex hull.

This is the exercise from J.Conway's textbook of complex analysis.Looking forward to any kind of help.


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