Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of holomorphic functions $f_n:U\to \mathbb{C}$. Suppose this sequence is Cauchy with respect to the $L^2$ norm. Then $\{f_n\}$ converges uniformly on compact subsets of $U$.
I have proved the following lemma.
Let $0<s<R$, and assume $f$ is holomorphic on $\overline{D}(0,R)$. Then there exists constants $C_1$ and $C_2$ such that $$ \|f\|_s\leq C_1\|f\|_{1,R}\leq C_2\|f\|_{2,R} $$ whereby $\|\cdot\|_s$ is the $\sup$-norm on $\overline{D}(0,s)$ and $\|\cdot\|_{i,R}$ the $L^i$-norm on $\overline{D}(0,R)$ for $i=1,2$.
Now, I want to use this lemma to prove my statement.
Let $V$ be a compact subset of $U$. I want to cover $V$ by a finite number of closed disks $\overline{D_i}(R_i,z_i)$ which I can make a little bigger such that they are still contained in $U$. How can I express this more foramlly? Then, if $\{f_n\}$ converges uniformly on each disk $\overline{D_i}$ does it converge uniformly on $U$?