I want to prove that $\prod f_n (z)$ uniformly converges for every compact subset of complex plane.
The textbook goes like this: It is enough to show that we can find a convergent series $ \sum M_n$ $ |Log f_n (z)| \le M_n$ on each $|z| \le R$. But why is it enough?
Or more generally, is there something like this: $\prod f_n (z)$ uniformly converges on $|z| \le R$ iff $\sum Log f_n (z)$ uniformly converges on $|z| \le R$ (or some other $R'$)?