Lemma 3

Let $X_1$ and $X_2$ be compact subsets of $\mathbf{C}$. If $X_2$ is contained in $X_1$ and if the boundary of $X_1$ in $\mathbf{C}$ is contained in $X_2$, then $X_1$ is the union of $X_2$ and of certain bounded connected components of the complement of $X_2$ in $\mathbf{C}$.

Let U be a connected component of $\mathbf{C}-X_2$. Every boundary point of $X_1\cap U$ in the open set U is also a boundary point of $X_1$ in $\mathbf{C}$, hence belongs to $X_2$ by hypothesis; since $U\cap X_2=\emptyset$, one sees that $X_1\cap U$ has no boundary point in the space U. Since U is connected, the intersection $X_1\cap U$ is either empty or equal to U (TG, I, p. 82, cor.), and the lemma results.