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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.

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1 Answer 1

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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.

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