1) it's not a proof by contradiction, it's a proof of a negation :grump:
2) I am not a fan of this phrasing "we can't simplify forever". Why can't we? It's obvious if you phrase it in the usual way as "the denominator is strictly smaller than it was before", but the "simplify" operation is kind of complex! 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).
2) I am not a fan of this phrasing "we can't simplify forever". Why can't we? It's obvious if you phrase it in the usual way as "the denominator is strictly smaller than it was before", but the "simplify" operation is kind of complex! 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).