What exactly is your objection, it looks OK to me.My Complex Analysis book states the following interesting theorem:
Let be a open disk in . Let be a complex function whose domain contains the open disk. Let . If the parital derivaties all exist and are continous on and at the Cauchy-Riemann equations are satisfied then is differenciable at .
This is a partial converse for the necessay Cauchy-Riemann condition.
Here is how the proof begins.
Let and let be so that (that is they lie in the open disk).
By the Mean Value Theorem,
And it goes on....
My problem is that, they did not use the MVT properly. Do you agree? (The problem is that this is one of those books which likes to skip steps).