I'm working on a problem in chapter 10 of Rudin's Real and Complex Analysis. The chapter focuses on Holomorphic functions and the problem is the following:
Here is how I've started: Let $K$ be a compact subset of $\Omega$. If $\gamma$ is any circle in $\Omega$ then $$ \vert f_n(z) - f(z) \vert \le \vert f_n(z) - f_m(z) \vert \le \frac{1}{2\pi}\int_\gamma \frac{\vert f_n(w) - f_m(w) \vert}{\vert w-z \vert}dw$$ for any $z$ inside the circle. Since $K$ is compact, it can be covered by a finite number of such circles, and so if I could show that the integral goes to $0$ as $n \to 0$, then I'm basically done after I take the greatest $n$ from all the circles. I assume this is where the DCT comes in, but I'm a little confused on how to apply it here. I understand that $\vert f_n(z) - f_m(z) \vert$ is bounded on $K$ and that it approaches $0$ in $K$, but it doesn't seem clear to me that $\frac{\vert f_n(z) - f_m(z) \vert}{\vert w-z \vert}$ is bounded on $K$, let alone that it approaches 0.
Any elaboration on how to apply the DCT in this case would be much appreciated.