# Understanding the proof of convergence criterion for infinite products via the relation of series?

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.