In the text "Functions of One Complex Variable" by Steven G. Krantz and Robert E. Greene on (Pg.260) I'm having trouble understanding the proof to $\text{Theorem (8.1.7)}$ specifically how the author constructed the bounds in $(2)$ and $(3)$ utilizing $\text{Lemma (8.1.6)}$ ?
$\text{Theorem (8.1.7)}$
If the infinite product $$\prod_{j=1}^{\infty}(1+|a_{j}|)$$
converges so then
$$\prod_{j=1}^{\infty}(1+a_{j})$$
$\text{Lemma (8.1.6)}$
Let $a_{j} \in \mathbb{C}$ Set
$$P_{N}=\prod_{j=1}^{N}(1+a_{j}), \, \, \, \, \, \, \, \, \, \overline P_{N}=\prod_{j=1}^{N}(1+|a_{j}|)$$
Then $|P_{N}-1| \leq \overline P_{N}-1$
$\text{Proof of Theorem (8.1.7)}$
For some $N_{0}$ it holds that $a_{j} \neq 1$ if $j > N_{0}$. One can write for $J > N_{0}$, in $(1)$
$(1)$ $$Q_{j}=\prod_{j=N_{0}+1}^{J}(1+a_{j}) \, \, \text{and} \, \, \overline Q_{j}=\prod_{j=N_{0}+1}^{J}(1+|a_{j}|).$$
If $M > N > N_{0}$, then in $(2)$
$(2)$
$$|Q_{M}-Q_{N}| = |Q_{N}| \cdot \bigg | \prod_{j=N+1}^{M}(1 + |a_{j}|)) \leq |Q_{N}| \bigg | \prod_{j = N+1}^{M}(1+|a_{j}| - 1 \bigg |.$$
Utilizing $\text{Lemma (8.1.6)}$, I made the conclusion in $(3)$
$(3)$
$$|\overline Q_{M}- \overline Q_{N}| = |\overline Q_{N}| \cdot \bigg | \prod_{j=N+1}^{M}(1 + |a_{j}|) \leq |\overline Q_{N}| \bigg | \prod_{j = N+1}^{M}(1+a_{j}) - 1 )\bigg |.$$
You will have \begin{align*} |Q_{M}-Q_{N}|& = \bigg | Q_{N}\Big(\prod_{j=N+1}^{M}(1 + a_{j})-1\Big) \bigg | \\ & = |Q_{N}| \bigg | \prod_{j=N+1}^{M}(1 + a_{j})-1 \bigg |\\ &\leq |Q_{N}| \bigg | \prod_{j = N+1}^{M}(1+|a_{j}|) - 1 \bigg |\tag{using the lemma} \end{align*}