assume there exist and number N such that for all . This means that is analytic and bounded on , thus constant by Liouville. Thus for all .
Let be a complex power series that converges uniformly on .
Show that there is such that for all .
and I'm kind of impressed with myself for learning all the latex to write this out
Ok so I'm thinking that without loss of generality I can let be 0
then I'm pretty sure I'd have to use the definition of uniform convergences to create a contradiction
but I don't know what to do for this
my first guess was to say that suppose
then do something with the inequality
by using
I think I might be close, but I don't know what to do from here