Combination Theorem for Complex Derivatives/Sum Rule/Proof 1

Theorem
Let $D$ be an open subset of the set of complex numbers $\C$.

Let $f, g: D \to \C$ be complex-differentiable functions on $D$

Then $f + g$ is complex-differentiable in $D$, and its derivative $\paren {f + g}'$ is defined by:


 * $\map {\paren {f + g}'} z = \map {f'} z + \map {g'} z$

for all $z \in D$.

Proof
Let $z_0 \in D$ be a point in $D$.

Define $k : D \to \Z$ by $\map k {z_0} = \map f {z_0} + \map g {z_0}$.

Then: