The main theorem is false: the paper's version of L'Hôpital's Rule is true in every ordered field, and the proofs of two key implications are circular or rely on an unstated stronger rule.
Toward a more complete list of completeness axioms
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L'H\^{o}pital's Rule is Equivalent to the Least Upper Bound Property
The main theorem is false: the paper's version of L'Hôpital's Rule is true in every ordered field, and the proofs of two key implications are circular or rely on an unstated stronger rule.