Let $(f_n(z))$ be sequence of complex functions.
By definition, infinite product $\prod f_n$ converges uniformly if there exists $N\in \mathbb{N}$ such that $F_N(z):=\lim_{m\to \infty} \prod_{n=N}^{m} f_n(z)$ is uniformly convergent and $F_N(z)$ has no zeros.
I want to show that uniform convergence of infinite product $\prod f_n$ is equivalent to the following two conditions:
(1)$f_n(z)$ converges uniformly to 1 as $n\to \infty$;this implies that, for sufficiently large n, the principal branch of the logarithm of $f_n$ can be defined.
(2)For sufficiently large n, $\sum \log f_n$ is uniformly convergent series.
Is this claim correct?If so, how can I prove it?
The proof is possible in the case of a sequence of complex numbers instead of a sequence of functions.