The question: Suppose $F(z) = \prod_{j = 1}^{\infty} [1 + f_{j}(z)]$ converges uniformily on compact subsets of a open set $U \subseteq \mathbb{C}$ to a limit that could be zero or non-zero. Suppose that $F(c) = 0$ for some $c \in U$. Prove or disprove: there is a $n \in \mathbb{N}$ such that $1 + f_{j}(c) = 0$.
Here's my work: I do think this is false but I am having trouble coming up with a normally convergent sequence of functions. Let $U = D(0, 1/2)-\{ {0} \}$ and $f_{j}(z) = \frac{z}{j^2}$. I want to try to examine $F(-1/2)$. Is this a good example? Thank you very much!