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@thewordninja_bk When you use Dedekind cuts, order and completeness come for free. It's definitely my bias as a general topologist (since I don't care as much about algebraic properties) but that's why I've never cared for Cauchy sequence constructions.
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@thewordninja_bk x≤y is literally x⊆y, and the least upper bound of A is ∪A. Can these be reimagined in terms of Cauchy sequences? Or maybe it's worth showing this alternative characterization?