A detail in the proof of Blaschke's condition via Jensen's formula

Question

I am trying to solve the following problem from Stein's complex analysis.

Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be the non-zero zeros of $\mathbb{f}$ in $\mathbb{D}$ counted according to multiplicity. Show that $$ \sum_{n=1}^\infty \left( 1 - \left|a_n \right|\right)\lt\infty $$

I have found a solution in MSE, as in If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$ and A holomorphic function with infinitely many zeros in the unit disc, and following the lines I have proved that $\Pi_{n=1}^\infty|a_n|$ converges, but I can't see why this impies $ \sum_{n=1}^\infty \left( 1 - \left|a_n \right|\right)\lt\infty $, and have been stuck here for hours. Any help would be appreciated!

Answer 1

The proof is highlighted in this Wikipedia page. Essentially, if $\prod |a_n|$ converges to say $p$, then $$\log p = \sum \log |a_n| = \sum \log (1 + (|a_n|-1)),$$

and by limit comparison the convergence of this last series is the same as the convergence of $\sum (|a_n| - 1)$.