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

Alex Rusterholtz; Kyle Hambrook; Martin Grant

REVIEW 4 major objections 3 minor 8 references

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

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In any ordered field, L'Hôpital's Rule, Taylor's Theorem with Peano Remainder, and the Limit of Derivatives Property are each equivalent to the Least Upper Bound Property.

desk verdict The main equivalence claim collapses because the paper's L'Hôpital rule is a trivial consequence of differentiability, but the (d)=> (a) construction is genuinely interesting. read the letter →

arxiv 2505.23092 v1 pith:NMGYLFFN submitted 2025-05-29 math.CA

Martin Grant , Kyle Hambrook , Alex Rusterholtz This is my paper

classification math.CA MSC 12J1526A24

keywords orderedfieldsleastupperboundpropertyDedekindcompletenessL'Hôpital'sruleTaylor'stheoremwithPeanoremainderlimitofderivativesnon-Archimedeanaxioms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that three familiar calculus statements—L'Hôpital's Rule, Taylor's Theorem with Peano Remainder, and the Limit of Derivatives Property—each characterize the Least Upper Bound Property in an arbitrary ordered field. If correct, this means each of these theorems stands as a completeness axiom for the real numbers, and the previously needed extra assumption of Countable Cofinality is unnecessary. The argument proceeds by chaining implications so that all three properties lead to the Limit of Derivatives Property, and then showing that any incomplete ordered field—Archimedean or not—fails that property through an explicit counterexample. The result matters because it ties ordinary differentiation theorems to the foundational question of what exactly makes the real line complete.

What carries the argument

The central object is a locally constant step function built from a partition of an incomplete ordered field $F$. In the Archimedean case, $F$ is a proper subfield of $\mathbb{R}$; one chooses $c\in\mathbb{R}\setminus F$, defines consecutive intervals $I_n=\{x\in F: c_n<|x|<c_{n-1}\}$ with $c_n=2^{-n}c_0$, picks a representative $a_n\in I_n$, and sets $f(t)=a_n$ for $t\in I_n$, $f(0)=0$. In the non-Archimedean case, the partition is instead given by Archimedean classes under $p\sim q$, meaning each of $|p|$ and $|q|$ is bounded by an integer multiple of the other; one representative is chosen from each nonzero class and $f$ is constant on each class. Because the pieces are open and $f$ is constant on them, $f'(t)=0$ for all $t\neq0$; because each representative sits at controlled distance from $0$, the ratio $|f(t)|/|t|$ stays between fixed positive constants. The construction is the load-bearing object behind the implication that incomplete fields violate the Limit of Derivatives Property.

What would settle it

Take an incomplete ordered field, for example the ordered field of rational functions with $x$ larger than every rational, and check the paper's stated L'Hôpital's Rule directly for differentiable $f,g$ with $f(a)=g(a)=0$ and $g'(a)\neq0$. The expansions $f(x)=f'(a)(x-a)+o(x-a)$ and $g(x)=g'(a)(x-a)+o(x-a)$ follow from the definition of differentiability alone, so $\lim_{x\to a} f(x)/g(x)=f'(a)/g'(a)$ holds without any completeness property; if that is the case, L'Hôpital's Rule cannot be equivalent to the Least Upper Bound Property.

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Extended reading notes

Core claim

The central claim of the paper is Theorem 3: in an ordered field $F$, the Least Upper Bound Property is equivalent to each of (b) L'Hôpital's Rule, (c) Taylor's Theorem with Peano Remainder, and (d) the Limit of Derivatives Property. The proof's main work is the direction (d) implies (a), which is handled by contrapositive: if $F$ is incomplete, the paper constructs a function $f$ that is continuous at $0$, has derivative $0$ at every nonzero point, and satisfies $\lim_{t\to 0} f'(t)=0$, yet whose difference quotient $f(t)/t$ does not tend to $0$ at $0$. Consequently the derivative of $f$ at $0$ cannot equal $\lim_{t\to0} f'(t)$, so the Limit of Derivatives Property fails. The same failure is produced in the Archimedean case by viewing $F$ as a proper subfield of $\mathbb{R}$ and slicing the field into open intervals, and in the non-Archimedean case by using the equivalence classes of the relation $p\sim q$ (comparability up to fixed integer multiples).

Load-bearing premise

The load-bearing step is the implication (b) $\Rightarrow$ (d), where L'Hôpital's Rule is applied to $F(x)=f(x)-f(a)$ and $G(x)=x-a$ to prove that $f$ is differentiable at $a$; this application already assumes differentiability of $f$ at $a$, which is precisely the conclusion the Limit of Derivatives Property is meant to deliver.

Editorial extensions

If this is right

If the theorem is correct, each of the three statements can serve as a definition of completeness for ordered fields, in place of the Least Upper Bound Property.
L'Hôpital's Rule becomes a completeness characterisation without needing the Countable Cofinality condition that earlier work required.
Taylor's Theorem with Peano Remainder is shown to be completeness-characterising even though its $n=1$ case is true in every ordered field.
The Limit of Derivatives Property emerges as the connecting mechanism: the other equivalences route through it, and it is the one that fails on incomplete fields.
First-course proofs that the real numbers satisfy L'Hôpital's Rule, Taylor's Theorem, and the derivative-limit theorem can all be re-read as consequences of completeness alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable extension is to ask whether the standard form of L'Hôpital's Rule, which assumes differentiability on a punctured neighborhood rather than only at the point $a$, is also equivalent to completeness; the current argument does not address that version.
The central role of the Limit of Derivatives Property suggests screening other one-point derivative identities as candidate completeness axioms, such as a rule for differentiating under the limit sign or a local monotonicity statement.
The non-Archimedean construction shows that a single point can carry a failure of the derivative-limit property while the field is otherwise rich; this hints that other pointwise 'continuity of derivative' statements may detect order completeness in a similar way.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline is that this paper has a load-bearing flaw, and it's not subtle: the L'Hôpital rule they define in Section 2 is just the algebra of limits applied to the difference quotients. If f and g are differentiable at a, with f(a)=f'(a)=g(a)=0 and g'(a)≠0, then f(x)/g(x) equals the quotient of the two difference quotients, and since each converges to a finite limit with denominator nonzero, the conclusion follows in any ordered field. So property (b) holds in Q, in the Laurent series field, everywhere. It cannot be equivalent to the Least Upper Bound Property. The stress-test note is right, and the reader's verdict of REJECT is correct.

What the paper does well is the (d)⇒(a) direction. That is a real construction: starting from an incomplete Archimedean field, they build a locally constant function with a puncture-neighborhood derivative limit of 0 but with f'(0) ≠ 0, and they adapt the idea to the non-Archimedean case with Archimedean classes. That half is substantial and, as far as I can tell, correct. It deserves to survive as a standalone result, though it only proves that the Limit of Derivatives Property implies completeness — which is consistent with the literature and not the advertised equivalence.

The soft spots are the rest of the proof chain. (b)⇒(d) is circular: applying L'Hôpital to F(x)=f(x)-f(a) and G(x)=x-a requires F to be differentiable at a, which is exactly what the Limit of Derivatives Property is trying to establish. (b)⇒(c) silently uses a stronger, unstated L'Hôpital rule that would require repeated differentiability and a higher-order version the paper never formulates. Since (b) is true everywhere, those implications cannot be doing any work; the only content is (c)⇒(b), which is the n=1 Peano case and is also true everywhere, as the authors themselves note. So the equivalence triangle collapses.

Who is this paper for? Someone working in 'real analysis in reverse' might mine the (d)⇒(a) construction and the Archimedean-class technique. But the advertised result is false, and the proof has textbook-level circularity. I would not cite the equivalence claim, and I would not send this to a serious referee without major revision — the authors should recast it as a note on the Limit of Derivatives Property and drop the L'Hôpital equivalence entirely. That said, the construction in the second half is worth engaging with, so I'd give it a skeptical but genuine look at a reading group.

Recommendation: reject the current version, but invite a rewrite that isolates the (d)⇒(a) result.

Referee Report

4 major / 3 minor

Summary. The paper claims to prove that, in an arbitrary ordered field, the Least Upper Bound Property is equivalent to each of three analytic properties: L'Hôpital's Rule as defined in Section 2, Taylor's Theorem with Peano Remainder, and the Limit of Derivatives Property. The proof strategy is to establish the implications (a)⇒(b), (b)⇒(c), (c)⇒(b), (b)⇒(d), and (d)⇒(a), with the last implication split into an Archimedean and a non-Archimedean case.

Significance. If the main theorem were correct, it would improve on Deveau and Teismann's result by removing the Countable Cofinality assumption from the L'Hôpital characterization of Dedekind completeness, and it would add two new completeness characterizations. The construction in the (d)⇒(a) direction, using step functions on Archimedean classes, is genuinely inventive and could be a useful contribution to the literature on completeness axioms for ordered fields. However, the central equivalence is false as stated: the paper's L'Hôpital's Rule is a trivial consequence of differentiability and holds in every ordered field, including incomplete fields such as Q. The paper therefore does not establish what it claims, and the main theorem cannot be repaired by minor edits within the current definitional framework.

major comments (4)

[§2, Definition of L'Hôpital's Rule; §3, Theorem] The stated property (b) is true in every ordered field. For any functions f and g differentiable at a with f(a)=g(a)=0 and g'(a)≠0, the identity f(x)/g(x) = [(f(x)-f(a))/(x-a)] / [(g(x)-g(a))/(x-a)] holds for x≠a. In any ordered field, the numerator tends to f'(a), the denominator tends to g'(a)≠0, and the quotient of two convergent limits with nonzero denominator is the quotient of the limits. Hence every ordered field, including the incomplete field Q, satisfies (b). Since Q does not satisfy the Least Upper Bound Property, the implication (b)⇒(a) cannot hold, and the main theorem is false as stated.
[§3, proof of (b)⇒(d)] The proof applies L'Hôpital's Rule to F(x)=f(x)-f(a) and G(x)=x-a. The Limit of Derivatives Property assumes only that f'(x) exists on a punctured neighborhood of a and that lim f'(x) exists; it does not assume f is differentiable at a. But applying the paper's rule requires F to be differentiable at a, which is exactly the conclusion that f is differentiable at a. The argument therefore assumes the differentiability it is meant to establish and is circular.
[§3, proof of (b)⇒(c)] For n>1, the function G(x)=(x-a)^n has G'(a)=0, so the first-order rule (b), which requires g'(a)≠0, cannot be applied directly to F/G. A repeated application would require a generalized L'Hôpital rule covering the case where f and g, together with several derivatives, vanish at a and the first nonvanishing derivative of g is of order n. No such generalized rule is stated, and it cannot be derived from (b), which is the trivial first-order identity holding in every ordered field. Thus the implication (b)⇒(c) is not established.
[§3, proof of (d)⇒(a), both cases] The displayed equalities f'(0)=lim_{t→0} f(t)/t ≠ 0 assert that the limit exists. The preceding estimates only give uniform bounds of the form 1/2<|f(t)/t|<2 (Archimedean case) or 1/C<|f(t)/t|<C (non-Archimedean case). These bounds rule out convergence to 0 but do not imply that the limit exists. If the limit fails to exist, the construction still refutes the Limit of Derivatives Property, but the proof must say so explicitly; as written, the equality is unjustified.

minor comments (3)

[Throughout] The text contains repeated misspellings: 'Archimedian' and 'Archimdean' should be 'Archimedean', and 'Least Upper Bounded Property' in the introduction should be 'Least Upper Bound Property'.
[§2, Taylor's Theorem with Lagrange Remainder] The displayed formula has a formatting error: the summation and the preceding term are not properly aligned. Please correct the typesetting of the displayed equation.
[§2, Remarks (2)] The proof that the Mean Value Theorem implies the Least Upper Bound Property uses the statement that both A∪(−∞,a) and its complement are open. This is plausible for a subset with no least upper bound, but the openness of the complement deserves a brief justification, since the argument is not immediate for an arbitrary subset A.

Circularity Check

2 steps flagged · score 8.0 of 10

The central implications (b)⇒(d) and (b)⇒(c) each use L'Hôpital's Rule in a way that presupposes the conclusion: (b)⇒(d) requires differentiability at a to prove differentiability at a, and (b)⇒(c) needs an unstated generalized L'Hôpital rule whose conclusion is exactly Taylor's Peano Remainder.

self definitional [Section 3, proof of Theorem, implication (b) ⇒ (d)]
"(b) ⇒ (d): Apply L’Hôpital’s Rule to evaluate f′(a) = lim x→a F(x)/G(x), where F(x) = f(x) − f(a) and G(x) = x − a."

The Limit of Derivatives Property requires proving that f is differentiable at a. But the paper's L'Hôpital's Rule, defined in Section 2, begins with the hypothesis that 'f, g : F → F are differentiable at a'. Applied to F = f − f(a), this hypothesis is exactly that f is differentiable at a — the very conclusion to be established. Moreover, the rule also requires F′(a) = 0, which would force f′(a) = 0 in general, and the concluding limit lim F/G is by definition f′(a). Thus the step assumes the existence of f′(a) in order to derive it; the derivation reduces to the conclusion by construction.
other [Section 3, proof of Theorem, implication (b) ⇒ (c)]
"(b) ⇒ (c): Apply L’Hôpital’s Rule repeatedly to evaluate limx→a F(x)/G(x), where F(x) = f(x) − Σ_{k=0}^n f^{(k)}(a)/k! (x−a)^k and G(x) = (x−a)^n."

The stated L'Hôpital's Rule requires g′(a) ≠ 0, but for G(x) = (x−a)^n with n ≥ 2, G′(a) = 0, so the rule cannot be applied even once. A repeated application would require a generalized L'Hôpital rule asserting that if F and G have vanishing derivatives through order n−1 at a and G^{(n)}(a) ≠ 0, then lim F/G = F^{(n)}(a)/G^{(n)}(a). For this choice of G and for F equal to f minus its nth Taylor polynomial, that generalized rule gives exactly lim F/G = 0, which is precisely Taylor's Theorem with Peano Remainder. Hence the proof substitutes the target statement as an unstated premise; (c) is not derived from (b) as defined.

full rationale

The paper's central theorem is that L'Hôpital's Rule, Taylor's Theorem with Peano Remainder, and the Limit of Derivatives Property are each equivalent to the Least Upper Bound Property in an arbitrary ordered field. The only substantive forward implications are (b)⇒(c) and (b)⇒(d); the remaining implications are either textbook (a)⇒(b), immediate from the n=1 Taylor Peano case (c)⇒(b), or the separate construction (d)⇒(a). Both forward implications fail as stated, and in each case the failure is a circularity rather than merely a missing detail. For (b)⇒(d), the quoted proof applies L'Hôpital's Rule to F(x)=f(x)−f(a) and G(x)=x−a in order to 'evaluate' f′(a); but the rule's hypotheses include differentiability at a, which is precisely the conclusion of the Limit of Derivatives Property. For (b)⇒(c), the proof says 'apply repeatedly', yet the stated rule cannot be used even once because G′(a)=0 for n≥2; the only way to make the repeated application work is to assume a generalized L'Hôpital rule whose conclusion for this particular F and G is exactly the Peano remainder statement. Because the paper's own definition makes L'Hôpital's Rule a trivial consequence of differentiability that holds in every ordered field, the claimed implication (b)⇒(a) cannot be correct; the derivation chain contains steps that reduce to their own conclusions by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities appear. The proof relies on standard theorems about ordered fields, but the fatal flaw is a logical gap in the implication chain, not an unstated axiom.

assumptions (3)

standard math Every Archimedean ordered field is isomorphic to a subfield of R.
Used in Case 1 of the proof of (d) ⇒ (a) to choose c ∈ R \ F and construct the intervals I_n.
standard math Archimedean classes partition a non-Archimedean ordered field and each non-zero class is open.
Used in Case 2 to define a locally constant function on equivalence classes.
standard math The Mean Value Theorem is equivalent to the Least Upper Bound Property in ordered fields.
Cited in the introduction and Remarks to motivate the equivalence program; not directly used in the main proof.

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Cite this review

Pith. "Pith review of L'H\^{o}pital's Rule is Equivalent to the Least Upper Bound Property." pith.science (2026). https://pith.science/paper/NMGYLFFN

@misc{pith2026250523092,
  author       = {Pith},
  title        = {Pith review of: L'H\^opital's Rule is Equivalent to the Least Upper Bound Property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMGYLFFN}},
  note         = {Machine review of arXiv:2505.23092}
}

read the original 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.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Robert G. Bartle. The elements of real analysis. John Wiley & Sons, New York-London- Sydney, second edition, 1976

work page 1976
[2]

72 + 42: characterizations of the completeness and Archimedean properties of ordered fields

Michael Deveau and Holger Teismann. 72 + 42: characterizations of the completeness and Archimedean properties of ordered fields. Real Anal. Exchange, 39(2):261–303, 2013/14

work page 2013
[3]

¨Uber die nichtarchimedischen Gr¨ oßensysteme

Hans Hahn. ¨Uber die nichtarchimedischen Gr¨ oßensysteme. Sitzungsber., Abt. IIa, ¨Osterr. Akad. Wiss., Math.-Naturwiss. Kl., 116:601–655, 1907

work page 1907
[4]

T. W. K¨ orner. A companion to analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004. A second first and first second course in analysis

work page 2004
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Real analysis in reverse

James Propp. Real analysis in reverse. Amer. Math. Monthly, 120(5):392–408, 2013

work page 2013
[6]

Principles of mathematical analysis

Walter Rudin. Principles of mathematical analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-D¨ usseldorf, third edition, 1976. 6 MARTIN GRANT, KYLE HAMBROOK, ALEX RUSTERHOLTZ

work page 1976
[7]

On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers

Khodr Shamseddine and Todd Sierens. On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers. ISRN Math. Anal., pages Art. ID 387053, 20, 2012

work page 2012
[8]

Toward a more complete list of completeness axioms

Holger Teismann. Toward a more complete list of completeness axioms. Amer. Math. Monthly, 120(2):99–114, 2013

work page 2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Robert G. Bartle. The elements of real analysis. John Wiley & Sons, New York-London- Sydney, second edition, 1976

work page 1976

[2] [2]

72 + 42: characterizations of the completeness and Archimedean properties of ordered fields

Michael Deveau and Holger Teismann. 72 + 42: characterizations of the completeness and Archimedean properties of ordered fields. Real Anal. Exchange, 39(2):261–303, 2013/14

work page 2013

[3] [3]

¨Uber die nichtarchimedischen Gr¨ oßensysteme

Hans Hahn. ¨Uber die nichtarchimedischen Gr¨ oßensysteme. Sitzungsber., Abt. IIa, ¨Osterr. Akad. Wiss., Math.-Naturwiss. Kl., 116:601–655, 1907

work page 1907

[4] [4]

T. W. K¨ orner. A companion to analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004. A second first and first second course in analysis

work page 2004

[5] [5]

Real analysis in reverse

James Propp. Real analysis in reverse. Amer. Math. Monthly, 120(5):392–408, 2013

work page 2013

[6] [6]

Principles of mathematical analysis

Walter Rudin. Principles of mathematical analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-D¨ usseldorf, third edition, 1976. 6 MARTIN GRANT, KYLE HAMBROOK, ALEX RUSTERHOLTZ

work page 1976

[7] [7]

On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers

Khodr Shamseddine and Todd Sierens. On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers. ISRN Math. Anal., pages Art. ID 387053, 20, 2012

work page 2012

[8] [8]

Toward a more complete list of completeness axioms

Holger Teismann. Toward a more complete list of completeness axioms. Amer. Math. Monthly, 120(2):99–114, 2013

work page 2013