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I'm trying to prove the following fact

If $K$ is a compact set whose complement is not connected, then there exists a function $f$ holomorphic in a neighborhood of $K$ which cannot be approximated uniformly by polynomials in $K$.

This is an exercise from Stein & Shakarchi's complex analysis, and the book gives the following hint

Pick a point $z_0$ in a bounded component of $K^{c}$, and let $f(z)=\displaystyle\frac{1}{(z-z_0)}$. If $f$ can be approximated uniformly by polynomials on $K$, show that there exists a polinomial $p$ such that $\left |{(z-z_0)p(z)-1}\right |<1$. Use the maximum modulus principle to show that this inequality continues to hold for all $z$ of $K^{c}$ that contains $z_0$.

I don't know how to begin the problem, any hint will be appreciated, please don't give answers.


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up vote 1 down vote accepted

You can begin by considering the case where $K=\{z\in \Bbb C:|z|=1\}$ and $z_0=0$, which is simpler but uses the same ideas. These ideas are described here.

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ok, i hadn't seen it right away because i hadn't seen that inmediate consequence to the maximum modulus principle, you need only to check that if $\Omega$ is the bounded component containing $z_0$, then $\bar{\Omega}-\Omega\subseteq{K}$. Thanks!!! – Camilo Arosemena Oct 14 '12 at 16:26

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