@MatheinBoulomenos Hey do you want to take a look at my proof?
I have one question though. I'm fine with the fact that if the limits of the real and imaginary part exists then the limit of their sum exists which is just the linearity property of the limit. But I'm not comfortable with the other direction, since for ex. $\lim 0 = \lim (x-x)$ exists, but $\lim x + \lim -x$ not?! So is there a problem with the proof?