prove that $f_n$ a sequence of holomorphic functions is a Normal family

Question

let $f_n=u_n+v_n$ be a sequence of holomorphic functions on $D(0,1)$ that are continuous on $ \overline {D(0,1)}$.Suppose that sequence $\{u_n\}$ converges uniformaly on $\partial D(0,1)$ and the sequence $\{f_n(0)\}$ converge.show that $\mathcal{F}=\{f_n|n=1,2,3,.....\}$ is normal family.

how to approach this problem. Do we need Montel theorem to prove this family is Normal ??? please help.Thanks in Advanced.

Answer 1

You should use the Schwartz integral formula (see this Wikipedia link). The sequence will converge uniformly on compact subsets of the disk.