In the text "Functions of one Complex Variable" I'm having trouble understanding the proof for convergence criteria of an infinite product via it's relation to infinite series as seen in Corollary $(8.1.4)$
$Corollary \, (8.1.3)$:
If $a_{j} \in \mathbb{C}$, $|a_{j}| < 1$ then the partial product $P_{n}$ for $$\prod_{j=1}^{\infty} (1+|a_{j}|)$$
satisfies: $$\exp(\frac{1}{2}\sum_{}^{}|a_{j}|) \leq P_{n}\leq \exp(\sum_{}^{}|a_{j}|). $$
$Corollary \, (8.1.4)$ If:
$$\sum_{}^{}|a_{j}| < \infty$$
then:
$$\prod_{j=1}^{\infty} (1+|a_{j}|)$$
converges.
I observed that the author directly applied the previous result in Corollary $(8.1.3)$ directly to $(8.1.4)$. This initially begins by allowing the series in $(9.1)$ to exist
$(9.1)$ $$\sum_{}^{}|a_{j}| = M$$ Initially from $(9.1)$ applying the following observations can be made:
$$\sum_{}^{}|a_{j}| = a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+ \cdot \cdot \cdot + a_{n}=M$$.
Now the partial product for $a_{j}$ can be defined as follows: $$\prod_{j=1}^{\infty} (1+|M|)$$
Our product satisfies the following inequality sated below:
$$\exp(\frac{1}{2}\sum_{}^{}|M|) \leq P_{n}\leq \exp(\sum_{}^{}|M|)$$.
The final result which the concludes the proof is the following:
$$P_{n} \leq \exp M$$
In summary my question is how did the inequality in Corollary $(8.1.4)$ was used to show that our infinite product converges, I'm missing any small but fundamental observations.
Note that $1 + |a_{n+1}| \geqslant 1$ for all $n$. Hence, $P_{n+1} = P_n(1 + |a_{n+1})\geqslant P_n$.
Since the sequence $(P_n)$ is nondecreasing and bounded above by $\exp(\sum_{n=1}^\infty |a_n|)$, it converges.