Let $\displaystyle \sum_{n=0}^\infty a_n(z-z_o)^n$ be a complex power series that converges uniformly on $\displaystyle \mathbb{C}$.

Show that there is $\displaystyle {N}\epsilon \mathbb{N} $ such that $\displaystyle a_n = 0$ for all $\displaystyle n>N $.

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 $\displaystyle z_o$ 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 $\displaystyle a_{N+1} \neq 0$

then do something with the inequality

$\displaystyle \mid S_N(x)-S(x) \mid < \varepsilon$

by using $\displaystyle S_{N+1}(x) = S_N(x) + a_{N+1}$

I think I might be close, but I don't know what to do from here