Let $a,b, z_0$ denote complex constants. Use the definition of a limit to show that $\lim_{z \to z_0} (az + b) = az_0 + b.$
Here is what I have done:
\begin{align*} |az + b - (az_0 + b)| &= |az - az_0 + b - b|\\ &= |a(z - z_0)|\\ &= |a||z - z_0|. \end{align*}
So for a positive number $\epsilon$,
$|az + b - (az_0 + b)| < \epsilon \text{ whenever } |a||z - z_0| < \epsilon$
or in other words $|az + b - (az_0 + b)| < \epsilon$ whenever $|z - z_0| < \delta$ where $\delta = \epsilon/|a|$.
Have I proved the statement correctly?