# If a seq of holomorphic function on an open set $\Omega$, $\{f_n\}$ coverges uniformly to a function $f$ on every compact set then so does $f_n'$

The whole theorem looks like-

Let, $$f_n:\Omega\to\Bbb{C}$$ be a sequence of holomorphic functions where $$\Omega$$ is open set. Suppose, $$\{f_n\}$$ converges uniformly to a function $$f$$ on every compact subset. Then $$f$$ is holomorphic on $$\Omega$$ and $$\{f_n'\}$$ converges uniformly to a function $$f'$$ on every compact subset.

I have proved the holomorphicity of $$f$$ using Morera's theorem. Now I try to prove the second part.
Let $$D=D_\delta(z_0)$$ be an open disc in $$\Omega$$ such that $$\overline {D}\subset\Omega$$
Let, $$F_n(z)=f_n(z)-f(z)\ \forall z\in \Omega$$, $$F_n$$ is holomorphic on $$\Omega$$.
Note that since $$\overline D$$ is compact, $$\operatorname{sup}_{z\in\overline D}|F_n(z)|$$ exists.
Choose $$\varepsilon>0$$, then $$\exists K\in\Bbb{N}$$ such that $$|f_n(z)-f(z)|<\delta\varepsilon\ \forall n\ge K, \ \forall z\in \overline D$$ (as $$\overline D$$ is compact) thus, $$\operatorname{sup}_{z\in\overline D}|F_n(z)|<\delta\varepsilon.$$
Now we apply Cauchy integral formula, $$F_n'(z)={1\over 2\pi i}\int_{\partial D}{F_n(\zeta)\over (\zeta-z_0)^2}d\zeta\implies |F_n'(z)|\le{1\over 2\pi}\operatorname{sup}_{z\in\partial D}|F_n(z)| 2\pi\delta{1\over \delta^2}\le\varepsilon\ \forall z\in \overline D\ \forall n\ge K$$.
Thus, $$|f_n'(z)-f'(z)|<\varepsilon\ \forall z\in \overline D\ \forall n\ge K$$.
Hence, $$f_n'$$ coverges uniformly to $$f'$$ on every compact disc inside $$\Omega$$.
Now, how to generalize the proof for any arbitrary compact disc? Thanks for assistance in advance.

For an arbitrary compact $$K\subset\Omega$$, we may find finitely many disks $$D_1,\cdots,D_n$$ which cover $$K$$ and whose closures are contained in $$\Omega$$. We may apply your proof separately to each disk $$D_i$$, giving us indices $$n_i$$ for which $$\sup_{D_i}|f'_n-f'|<\epsilon \forall n\geq n_i$$. Taking the maximum $$N:= \max_i n_i$$, we obtain that $$\sup_{K}|f'_n-f'|\leq \max_i \sup_{D_i}|f'_n-f'|<\epsilon \forall n\geq N$$ giving us locally uniform convergence as required.

• Ok, it's easy, got it. – Biswarup Saha Apr 13 '19 at 8:03