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