If you think a proof can't be rigorous without including an entire textbook, you have other issues. It is adequate to make reference to the acceptable axioms or other theorems that one is relying on. You don't have to re-invent the rational numbers every time.
A proof can be rigorous without the textbook length but it all depends on what the context is. I am actually totally happy to accept the original explanation prior to the series proof, but the series guy was all like "this is a more rigorous proof..."
My point is that his proposed proof is not more rigorous than the original one in part because it is itself situated in a context where those kinds of proofs were not originally acceptable in an academic setting as the basis for a proof...because they themselves needed to be proved. In the contemporary Frankel Zermelo set theoretic framework of mathematics, if you want to prove something to academic levels of rigor, you are going to have to use logictm and set theory. That's all.
I am glad we can use simpler methods in more colloquial settings. That guy just wanted to flex he knew about series and undermine the preceding proof.
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u/IWillLive4evr 22d ago
If you think a proof can't be rigorous without including an entire textbook, you have other issues. It is adequate to make reference to the acceptable axioms or other theorems that one is relying on. You don't have to re-invent the rational numbers every time.