Runge's theorem and polynomially convex hull 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. 
 A: Assume that $K\subseteq G$ is compact.
We show that (a)$\Rightarrow$(c)$\Rightarrow$(b). Suppose that (a) holds. Pick $z_0\in G-K$ but assume that $|f(z_0)|\leq \|f\|_K$ for every $f\in H(G)$. Note that since $K$ is bounded we can find $M>0$ such that $|z-z_0|\leq M$ for $z\in K$. Choose $g\in H(G)$ such that
    \begin{equation}
  \left|g(z)-\frac{1}{z-z_0}\right|< \frac{1}{M}.
 \end{equation}
    Then 
    \begin{equation*}
  |(z-z_0)g(z)-1|<1
 \end{equation*}
    on $K$. Since $h(z) =(z-z_0)g(z)-1\in H(G)$ we gather that $1 = |h(z_0)|<1$ which is a contradiction. Therefore (a) implies (c).
Assume that (c) holds. Let $D$ be a bounded component of $G-K$. Is $D^-\cap \partial G = \square$ then $\partial D\subseteq K$. But then we can not have the existence of a holormorphic function $f\in H(G)$ with the property $|f(z)|>\|f\|_K$ for some $z\in D$ by the maximum modulus principle. Therefore $D^-\cap\partial G\neq \square$. 
Assume that (b) holds. Then the previous exercise VIII.1.3 implies that every component of $C_\infty-K$ contains a component of $C_\infty-G$. Let $E$ be a set which meets every component of $C_\infty-G$. Then Runge's Theorem implies that we can approximate any holomorphic function on a neighborhood of $K$ by rationals functions with poles in $E$ arbitrarily well. Since these rational functions belong to $H(G)$ we are done.
