Uniform limit of holomorphic functions is holomorphic

Statement

Suppose $f_n$ is a sequence of holomorphic functions on an open subset $U\subset \mathbb{C}$, and $f_n\to f$ pointwise, with the convergence being uniform on compact subsets. Then $f:U\to \mathbb{C}$ is a holomorphic function, and moreover, for any $k$, we have:

$f_n^{(k)}\to f^{(k)}$

with this convergence also being uniform.

Facts used

• Goursat's integral lemma
• Morera's theorem
• Cauchy integral formula for derivatives

Proof

The first assertion follows directly from Goursat's integral lemma and Morera's theorem.

• Since each $f_n$ is holomorphic, it integrates to $0$ along triangles, by Goursat's lemma.
• Hence, $f$ integrates to 0 along triangles, by uniform convergence on compact subsets, and the exchange of limit and integral.
• Hence, $f$ is also holomorphic, by Morera's theorem.

The second assertion follows using the Cauchy integral formula for derivatives. This expresses the value of the derivative in temrs of an integral involving the function. Fill this in later