I came across another proof by chance in some lecture notes I was reading. I felt the appropriate thing was to copy (with citation!) that proof into another answer. Sure, it didn't help me, but MO is more than just "user asks question; user gets answer; bye bye". I wanted to share this for the benefit of the community.
I happen to look at the question again today, and I see that a moderator deleted my answer. They didn't comment on it, suggesting improvement. Just a drive-by delete. Putting aside the rudeness/disrespect/whatever-word-you-want*,
What was the reason for this deletion? and, why was there no communication?
Are self-answers no longer encouraged? Certainly I'm not encouraged to support the community given this treatment.
*It's fairly well established that this sort of behaviour is what has driven many people away from the Stack Exchanged sites. I used to be fairly active, with >5k rep gained via answers over a few years, but packed it in, barely posting questions or answers now, largely due to this sort of behaviour