I'm having difficulty showing that:
$\displaystyle \lim_{z \leftarrow z_o} \text{Re}\ (z) = \text{Re}\ (z_0)$
it just seems like an obvious
You have to show that for all $\displaystyle \varepsilon > 0$ there exists a $\displaystyle \delta>0$, such that whenever:
$\displaystyle |z-z_0|<\delta$
then
$\displaystyle | {\rm{Re}}(z) - {\rm{Re}}(z_0)|<\varepsilon$
(In this case $\displaystyle \delta=\varepsilon$ will do which is perhaps why this seems obvious)
RonL