Limits in the complex vectorspace

[eptheta]

Hi,
I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion.
Here's what I have been trying to get at:

$\lim_{z\to z_o} z^2+c = {z_o}^2 +c$

$|z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta$

$LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|$

$=|z-z_o||\overline{z+z_o}|$

$=|z-z_o||\bar{z}+\bar{z_o}|$

$=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|$

$=| |z|^2 -|z_o|^2 +2Im(zz_o) |$

$\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)$

But I can't get any further. I did this much thinking I could factor it to the square of delta, but that didn't work out because of the positive 2zzo term.If anyone can help me out here, it would be great. Thanks.

 

[tiny-tim]

hi eptheta!

$LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|$

stop there … the |z-z_o| part is easy,

so all you have to do is ensure that |z+z_o| is bounded