We just learned about infinite products in class. There's no textbook for the course so I am struggling with the following two basic problems.
Let $ a_n(z) = 1 + b_n(z), |b_n(z)| \leq \lambda < 1, z \in E $. Prove that
(1) $ \prod (1+|b_n|) $ converges uniformly on $ E $ if and only if $ \sum |b_n| $ does;
(2) If $ \prod (1+|b_n|) $ converges uniformly on $ E $, then $ \prod (1+b_n) $ does too.
Part (1) seems to be just taking logarithm, because $ \prod (1+|b_n|) $ converges uniformly is the equivalent to $ \sum \log (1+|b_n|) $ converges uniformly, but $ \log (1+|b_n|) \sim |b_n| $ as $ b_n \to 0 $. Is this correct or does it work only for pointwise convergence? I am not sure how to do part (2).
Any help is appreciated.