I'm looking for a detailed proof, or a reference to a detailed proof, of the following theorem.
Let $G$ be a non-empty, bounded domain in the Euclidean plane $\mathbb{R}^2$. Then $G$'s boundary is connected iff whenever $G$ contains a closed Jordan curve $\gamma$, $G$ also contains the interior of $\gamma$.
This is Theorem 4.14 on p. 70 of A. I. Markushevich's Theory of Functions of a Complex Variable, Three Volumes in One, 2nd Edition, Chelsea Publishing Company, 1977. Markushevich gives a detailed proof, however, one of the major steps of the proof is not adequately explained (it is the subject of this question of mine, which remains unanswered despite a bounty).
This post is essentially the same as my current post, however the answer provided there is not sufficiently detailed (as the writer of the answer admits freely in the beginning of the answer), and neither is a reference to an external source provided.