Linear Bound between Complex Function and Derivative

Theorem
Let $f: D \to \C$ be a complex-differentiable function, where $D \subseteq \C$ is an open set.

Let $z_0 \in D$.

Let $\epsilon \in \R_{>0}$.

Then there exists $r \in \R_{>0}$ such that for all $z \in \map {B_r} {z_0}$:


 * $\size {\map f z - \map f {z_0} - \map {f'} {z_0} \paren {z - z_0} } < \epsilon \size {z - z_0}$

where $\map {B_r} {z_0}$ denotes the open ball with center $z_0$ and radius $r$.

Proof
Given $\epsilon > 0$, we can find $r > 0$ by definition of limit such that for all $z \in \map {B_r} {z_0}$:


 * $\dfrac {\size {\map f z - \map f {z_0} - \map {f'} {z_0} \paren {z - z_0} } } {\size {z - z_0} } < \epsilon$

We rearrange to:


 * $\size {\map f z - \map f {z_0} - \map {f'} {z_0} \paren {z - z_0} } < \epsilon \size {z - z_0}$