I am going to present the proof of Weierstrass Factorization Theorem in a student seminar in the next week. For that purpose I am following Conway's book on complex analysis for the essential details required to prove the theorem. Here I came across a proposition regarding the convergence of an infinite product which I am unable to follow properly. Here it is $:$
$\textbf {Proposition} :$ Let $\text {Re}\ (z_n) \gt 0$ for all $n \geq 1.$ Then $\prod\limits_{n=1}^{\infty} z_n$ converges to a non-zero number iff the series $\sum\limits_{n = 1}^{\infty} \log z_n$ converges.
[Here "$\log$" indicates the principal branch of logarithm].
Proof. "$\impliedby$" part $:$ This direction is easier and I have understood the proof of this part quite well.
"$\implies$" part $:$ Let $p_n = (z_1 \cdots z_n),\ z = re^{i \theta}, - \pi \lt \theta \leq \pi,$ and $\ell (p_n) = \log |p_n| + i \theta_n$ where $\ \theta - \pi \lt \theta_n \leq \theta + \pi.$ If $s_n = \log z_1 + \cdots + \log z_n$ then $\exp\ (s_n) = p_n$ so that $s_n = \ell (p_n) + 2 \pi i k_n$ for some integer $k_n.$ Now suppose that $p_n \to z.$ Then $s_n - s_{n-1} = \log z_n \to 0;$ also $\ell (p_n) - \ell (p_{n-1}) \to 0.$ Hence, $(k_n - k_{n-1}) \to 0$ as $n \to \infty.$ Since each $k_n$ is an integer this gives that there is an $n_0$ and a $k$ such that $k_{m} = k_{n} = k\ $ for $\ m,\ n \geq n_0.$ So $s_n \to \ell (z) + 2 \pi i k;$ that is, the series $\sum \log z_n$ converges.
This completes the proof.
$\textbf {Question} :$ What is the reason behind defining the logarithmic branch $\ell$ on the slit plane $\mathbb C \setminus \{\alpha = \theta + \pi \}$ and considering the quantity $\ell (p_n)$ with respect to that branch? I would guess that $p_n$'s and $z$ are all lie in the slit plane $\mathbb C \setminus \{\alpha = \theta + \pi \}$ and the branch of logarithm $\ell$ is continuous (in fact, analytic) there. But I can't see why all $p_n$'s and $z$ lie in the slit plane. Could anyone please help me understanding this fact which is implicitly hidden in the proof?
Any help in this regard would be warmly appreciated. Thanks for investing your valuable time for reading my question.