> They don't even mention "decreasing" until the very final Note box where they say offhand that actually it's an infinite descent (which is a critical part of the proof they've otherwise handwaved).
That isn't actually a critical part of the proof; you can just assume that your initial two integers are relatively prime and then derive a contradiction directly.
Then why didn't they! Of course the proof can be fixed, we all know sqrt(2) is irrational, but why get so close to proving it and then just not finish the job?
That isn't actually a critical part of the proof; you can just assume that your initial two integers are relatively prime and then derive a contradiction directly.