Please undelete this answer:
sequences and series - Associativity of infinite products - Mathematics Stack Exchange
It is, of course, possible that it contains a logical error. In that case, someone should point out the nature and location of the error. That should not be difficult, because the argument is short and simple, even if incorrect!
The only comments so far are those by the OP, and my replies. As far as I can tell, his comments reiterate the central difficulty of the question, without ever addressing the argument given in my answer.
On two occasions, I asked him to explain what is the matter with my answer:
"Which of the logical steps in the last paragraph of my answer do you suspect may be invalid? (I'm sorry if I've made an obvious goof; but I just can't see it.)"
"Nevertheless, Theorem 2 does apply. If you maintain that it does not, please say which of its hypotheses are not satisfied."
He ignored both of these requests, so I stopped replying to his comments.
I am glad to see that the question has now received more attention. I hoped that if this happened, some third party would resolve the impasse over my answer, even if the resolution was not in my favour. But all that has happened is that my answer has been deleted, even though no-one but the OP has made a critical comment on it! It has also received an upvote, so at least one other person must believe that my argument is valid.
According to the Help Center:
Why and how are some answers deleted? - Help Center - Mathematics Stack Exchange
"Answer posts that do not fundamentally answer the question may be removed. This includes answers that are:
- commentary on the question or other answers
- asking another, different question
- "thanks!" or "me too!"-type responses
- exact duplicates of other answers
- barely more than a link to an external site (i.e. the actual answer is not included in the post)
- not even a partial answer to the actual question"
None of those reasons apply here. The only possible remaining justification is this:
"Moderators can delete any answer, and trusted community members can vote to delete answers that have a score of -1 or lower (3 votes will result in deletion)."
That is at best a technical justification. But it is no moral justification at all, in view of the lack of any critical comments apart from a repetitive sequence of comments by the OP, who is presumably also the one user who downvoted the answer. Whoever the downvoter was, the score was -1, the last time I looked before today.
Even that narrow technical justification no longer exists (if it ever did), because the answer score is now 0 (=-1+1).
I hope that even if the answer is eventually deleted again, someone by then will have given a reason!