Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $\mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero?
I was thinking about either: if f is everywhere nonzero then form $g = 1/f $ and then use $ |fg| = 1 $ to show that $ f $ is bounded, hence constant, or suppose that $ f $ is not constant, then show that $ f $ has a zero somehow. But I don't know how to do either. How do I prove this?
Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.