I want to know whether my attempt for the following proof is correct:
Let $(f_n)_{n\in \mathbb N}$ be a sequence of entire functions converging uniformly on $\partial D$ (boundary of the unit disk, i.e. unit circle) to some function $g: \partial D \to \mathbb C$. Show that there exists a holomorphic function $f: D \to \mathbb C$ such that $f_n(z) \to f(z)$ for each $z\in D$.
My attempt: By the Cauchy integral formula, the function $\tilde g: D \to \mathbb C$, $$z\mapsto \frac{1}{2\pi i}\int_{\partial D}\frac{g(\zeta)}{\zeta -z}d\zeta$$ is holomorphic on $D$. Now, for each $z\in D$:
$$\lim_{n\to \infty} f_n(z) = \lim_{n\to \infty} \frac{1}{2\pi i} \int_{\partial D}\frac{f_n(\zeta)}{\zeta -z}d\zeta = \frac{1}{2\pi i} \int_{\partial D}\lim_{n\to \infty} \frac{f_n(\zeta)}{\zeta -z}d\zeta = \frac{1}{2\pi i}\int_{\partial D}\frac{g(\zeta)}{\zeta -z}d\zeta$$ where we exchanged limit and integral by uniform convergence. So $\tilde g$ is our desired function.
By the maximum principle: $$ \sup_{z\in D} |f_n(z)-f_m(z)| \le \sup_{z\in \partial D} |f_n(z)-f_m(z)| $$ so $\{ f_n \}$ is uniformly Cauchy on $\bar D$. Hence $f_n$ converges uniformly to some (necessarily holomorphic on $D$ and continuous on $\bar D$) function $f$, where $f = g$ on $\partial D$.
The fact that $f$ is holomorphic on $D$ follows, for example, from Morera's theorem: If $\gamma$ is any closed curve in $D$, then $$ \int_\gamma f\,dz = \int_\gamma (\lim_{n\to\infty} f_n)\,dz = \lim_{n\to\infty} \int_\gamma f_n\,dz = 0 $$ by Cauchy's integral theorem and properties of uniform convergence. Hence $f$ is holomorphic.