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Aren't Dedekind cuts prior to Cauchy sequences? Dedekind cuts rely on ordering which seems to be less of an ask than Cauchy's "equivalence sets of sequences with this property". Or to put it differently, you can apply Cauchy to partially ordered sets, but not Dedekind.
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In other words it's a remarkable and non-obvious fact that "order-completing" the rationals and "metric-completing" the rationals gets you equivalent results.
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