If $a_n\ge 0$ and $a_n\to 0$ and also $\sum a_n=\infty$, then $\sum \left(1-\left|\frac{a_n -1}{a_n +1}\right|\right) = \infty$

Good Afternoon!

am trying to understand a well known research paper, I came across the following lines, which let me assume that the conjecture is pretty "standard" or well known, however, I don't know the name of the series and have no clue what to google to better understand them:

If there is a series $\{a_n\}$ with $a_n \geq 0$ and $\lim a_n \rightarrow 0$ and the series $\sum_{i=1}^{\infty}a_n = \infty$, then the series $\sum_{i=1}^{\infty} \left( 1 - \left|\frac{a_n -1}{a_n+1}\right|\right) = \infty$, diverges too.

Does anyone know the name of the series above? -- looks similar to Blaschke products?!Or any idea, in which book to look for some more explanation about it?

cheersJohn