Proof of Theorem 12.2.7 from Function Theory of One Complex Variable by Greene and Krantz

I am reading the book $textit{Function Theory of One Complex Variable}$ by Greene and Krantz. In Chapter 12 Rational Approximation Theory, Theorem 12.2.7 on p. 378 says that if $K$ is a compact set such that $widehat{mathbb{C}}-K$ has only finitely many connected components, then for function $f$ continuous on $K$ and analytic in the interior of $K$, it can be approximated by rational functions. They haven’t written the proof. Instead, they only said that “only the final argument in the proof of Theorem 12.2.1 (Mergelyan’s Theorem) needs to be modified so that $textbf{sets $E_j$ are selected in each component of $mathbb{C} -K$.}$” I don’t understand why the bold part is true. The construction of $E_j$ is given in the following:
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I have found many books on complex analysis and no book explains in details why Mergelyan’s Theorem can be extended to this general case. In fact, I am really frustrated with this. Can anyone help me?