L'H\^{o}pital's Rule is Equivalent to the Least Upper Bound Property

TL;DR

This paper establishes an equivalence between L'Hôpital's Rule and the Least Upper Bound Property in any ordered field, linking a fundamental calculus rule to a core property of ordered structures.

Contribution

It proves that L'Hôpital's Rule, Taylor's Theorem with Peano Remainder, and a related property are all equivalent to the Least Upper Bound Property in arbitrary ordered fields.

Findings

L'Hôpital's Rule holds if and only if the Least Upper Bound Property holds.

Taylor's Theorem with Peano Remainder is equivalent to the Least Upper Bound Property.

A third property related to L'Hôpital's Rule is also shown to be equivalent.

Abstract

We prove that, in an arbitrary ordered field, L'H\^{o}pital'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'H\^{o}pital's Rule.