Riemann Theorem

Question

In the book "Theory and Application of Infinite Series" by Knopp, the Riemann theorem for rearrangements of infinite products is cited on p.227, but not proven, i.e.

Let $\prod\limits_{n=1}^{\infty}(1+a_n)$ an infinite product which does not converge absolutely, no factor is zero of this product and $s\in \mathbb{R}$ is arbitrary.

1. There exist a rearrangement so that $\prod\limits_{n=1}^{\infty}(1+a_{\sigma(n)})=s$.
2. There exist a rearrangement so that $\prod\limits_{n=1}^{\infty}(1+a_{\sigma(n)})=+\infty$ or $\prod\limits_{n=1}^{\infty}(1+a_{\sigma(n)})=-\infty$.

I am really struggling with the proof. Can someone provide it?

Answer 1

Here it is, from your link...

Note the part about "same sign".

To prove it, we can remove the finitely many negative factors, then apply the series theorem to the logarithms of the remaining factors.