I really had to trouble myself to see how one can actually use the ...hint!
This result is easier to obtain with standard complex analysis arguments. And even if that is not satisfactory, there's always the Arzela-Ascoli theorem. But anyhow, can't show disrespect by ignoring advice.
Now for the proof. For a fixed , consider a circle around , the radius being so small as . Using the Cauchy integral formula,
and dominated convergence grants us
so that, for any such that
(Note: This is independent of ! Dominated Convergence rules )
Now we show , proving that the convergence is uniform. Using (1), forall we have
from which follows .
ps. I still think the non-Lebesgue argument is sweeter.