L'Hôpital's Rule is Equivalent to the Least Upper Bound Property

TL;DR

This paper shows that in any ordered field $ mathbb{F}$, the Least Upper Bound Property (LUBP) is equivalent to several fundamental theorems of analysis: L'Hôpital's Rule, Taylor's Theorem with Peano remainder, and the Limit of Derivatives Property. The key contribution is the chain of implications (a)⇔(b)⇔(c)⇔(d), including a nontrivial (d)⇒(a) argument that treats Archimedean and non-Archimedean cases to rule out the failure of completeness. The authors provide constructive arguments that L'Hôpital's Rule can characterize LUBP without extra assumptions, and they extend the equivalence to Taylor's Theorem with Peano remainder. These results deepen the understanding of which completeness-like properties are necessary to recover classical analysis results in abstract ordered fields, with implications for the landscape of equivalent formulations around real-analytic theorems. The work integrates notions such as Countable Cofinality and Archimedean classes to handle nonstandard ordered fields in the main reduction.

Abstract

We prove that, in an arbitrary ordered field, L'Hôpital's Rule is true if and only if the Least Upper Bound Property is true. We do the same for Taylor's Theorem with Peano Remainder, and for one other property sometimes given as a corollary of L'Hôpital's Rule.

Theorems & Definitions (2)

• Theorem
• proof