Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that
(i) $0<|a_n|<1$,
(ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$
Prove that the infinite product
$$\prod_1^\infty \frac{(a_n -z)|a_n|}{(1-\bar {a_n}z)a_n}$$
converges uniformly for $|z| \le r$ for any r <1, hence defining a holomorphic function in the unit disc.
I want to use the very useful theorem that $\prod_1^\infty (1+b_n)<\infty$ if and only if $\sum_1^\infty |b_n| < \infty$, but I am having some trouble with getting the factor in the form of $(1+b_n)$.
Also, assumption (ii) seems, at least on the surface, not useful / redundant, and that the primary goal is to get the $(1+a_n)$ form, so that we can immediately look at the sum of |an|.
Any hints on the problem -- or comments on how assumption (ii) is useful -- are greatly appreciated.
Thanks,