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John,
I've interspersed some comments below but first: Thank you so much for a useful and thoughtful answer. Your response shows that:
1. You read the original question carefully so you knew it was about x+y vs. 1/(1/x+1/y) rather than x+y vs. 1/x+1/y.
2. You comprehended the question so you knew it was about convincing people in a lasting manner rather than providing a proof.
3. You gave some thoughtful, thought-provoking, and potentially useful suggestions.
I am truly grateful.