The terminology of constructive real numbers is a giant pile of mess because subtly different concepts sometimes go by the same name or vice versa, and not every author takes the trouble to recall every relevant definition; but IIUC, the nLab page (and this question), the “Eudoxus reals” are the same as the (unmodulated?) Cauchy reals, which seems to be a more standard terminology. If this is correct, I suggest writing “Cauchy reals (aka Eudoxus reals)” in this question, unless you have some reason to emphasize Eudoxus (in which case, maybe state it).
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