Written: May 15, 2022.
ArXiv bouncers (that are euphemistically called moderators) (including Professor Victor Reiner (see below)) made the very poor decision to reject this gem.
First some background.
After it got rejected for the first time, I was sure that it was a quick oversight of a moderator who is very busy, and by taking a "quick look" decided that it is not "arXiv material" and rejected it. So I emailed Professor Victor Reiner, who is listed as a moderator, and whom I know personally, asking to intervene and reverse this poor decision. I was sure that he would appreciate this little piece (that says so much!), and support my appeal. To my shock, instead of an apology on behalf of the moderators who rejected the submission, and a quick reversal of the rejection, I got the following reply:
"There I agreed with a moderator's assessment that it didn't have enough new research content, since I think it is only your 5th proof that might count as "new"."
In making this poor decision, the moderators, including Professor Reiner, showed that while they may know everything there is to know about their particular tree, they have very limited vision of the mathematical forest.
Let me explain why.
First, suppose that indeed my little submission doesn't have significantly new "research content". What's wrong with exposition? It is often much more interesting and insightful than yet another technical theorem. Also the whole is more than the sum of its parts. Mark Levi's delightful gem reminded me of five other proofs, each belonging to five different undergraduate classes that I have taught during my long career. It shows the unity (and diversity!) of mathematics.
It can also be viewed as a "mathematical poem", but I guess that these arXiv bouncers (including Professor Reiner) have no appreciation of poetry.
But, it so happens, that the two last proofs, as far as I know, are new. If not, please, Professor Reiner, come up with references.
[Added June 24, 2022: Jose Hernandez Stgo pointed out that the fourth proof can be found in Spivak's "Calculus" ( see pages 313-314 of the fourth edition of the book)]
But, if nothing else, my little gem drew attention to my hero Mark Levi's even greater gem. This wonderful column deserves to be better known!