Suppose $\Omega\subset\mathbb{C}^n$ is a holomorphically convex domain and $\Delta\subset\mathbb{C}$ is the unit disc. Suppose $\varphi_i:\Delta\to \mathbb{C}$ is a series of continuous functions which satisfies $\varphi_i$ is holomorphic in $\Delta$. Prove that if $\cup_{i=1}^\infty \varphi_i(\partial\Delta)\subset\subset \Omega$, then $\cup_{i=1}^\infty \varphi_i(\Delta)\subset\subset \Omega$.
I just learned the concept of holomorphically convex domain and Cartan-Thullen Theorem. But I do not know whether it is useful here. If so, which equivalent condition should I use? And how to use the condition $\cup_{i=1}^\infty \varphi_i(\partial\Delta)\subset\subset \Omega$?
Let $K = \cup_{j=1}^\infty \varphi_j(\partial\Delta)$. Since $\Omega$ is assumed to be holomorphically convex, $\hat K \subset\subset \Omega$.
Assume $f$ is a holomorphic function on $\Omega$. Then for each $j$, $$\max_{z\in\varphi_j(\Delta)} |f(z)| \le \max_{z\in\varphi_j(\partial\Delta)} |f(z)| \le \max_{z\in K} |f(z)|$$ (by applying the maximum modulus theorem to $f \circ \varphi_j$). Hence $\hat K \supset \cup_{j=1}^\infty \varphi_j(\Delta)$ and we are done.