# Understanding the Archimedean Property

I've just started to learn real analysis and I'm confused about the meaning of the Archimedean Property because I found several different definitions and several terms that seem to be related but not at the same time.

In "Introduction to Real Analysis" by Bartle and Sherbert, the Archimedean Property is stated as follows:

"If x is any Real number then there exists n in the Natural numbers such that x<=n"

And this is supposed to mean that the Natural numbers are not bounded above in the Real numbers. I understand everything so far.

But then I found this definition online: "If x,y are any positive real numbers then there exists n in the Natural numbers such that nx>y"

I don't understand how this definition is the same as the one in the textbook? (1)

Also is this definition the more useful definition (because I looked at multiple websites and this is the definition they state rather than the one in my textbook)? (2)

Furthermore, the textbook states "Collectively, the Corollaries 2.4.4 - 2.4.6 are sometimes referred to as the Archimedean Property of the Real numbers":

2.4.4 "If S:={1/n: n in N}, then inf S = 0"

2.4.5 "If t>0, there exists n in N such that 0 < 1/n < t"

2.4.6 "If y>0, there exists n in N such that n-1 ≤ y < n"

[N stands for the Natural numbers]

I'm confused how these 3 Corollaries are related to the meaning of the Archimedean Property (3) and also when I search the Archimedean Property versus the Archimedean Property of the Real numbers, I get basically the same results? (4)

Thanks in advance for any help possible because I'm really confused and the more I search the more confused I get!

• When building the foundations of a subject you have to start somewhere. Quite often there are alternative places to start which turn out to be equivalent once they are understood. It is more important to understand how the corollaries are derived from the original statement of the property that to know which is the "proper" definition of the Archimedean property. Commented Nov 1, 2023 at 23:02
• Roughly, a set that has Archemidean property has no infinitely large nor infinitesimal numbers. All of the corollaries are making this point, either showing a construction that will generate a rational number smaller (closer to zero) than any real number or a natural number that is larger than any real number. Commented Nov 1, 2023 at 23:10
• Ok so 2.4.5 tells us there is no infinitesimal number because no matter how small you pick t to be, there is 1/n < t, but what does 2.4.4 and 2.4.6 tell us?
– vina
Commented Nov 2, 2023 at 5:57
• 2.4.6 is not correct. At least one of the two $<$ should be replaced by a $\le$. Commented Nov 2, 2023 at 21:33
• @AnneBauval yes
– vina
Commented Nov 5, 2023 at 16:08

I managed to get a copy of Bartle and Sherbert (third edition). Here on page 35, they define the infimum in Definition 2.3.2. The definition is only for nonempty subsets $$S$$ of $$\mathbb R$$. There is no definition for more general structures.

Their property 2.4.4 "If S:={1/n: n in N}, then inf S = 0" is a little odd, for the following reason. For the other properties, one can illustrate the non-triviality of the condition (i.e., the Archimedean property) by considering a proper ordered extension of $$\mathbb R$$ (for example, the Laurent series, or the hyperreals $$\mathbb R^\ast$$). There one clearly sees the failure of the Archimedean property, because if $$r>0$$ is real and $$n\in \mathbb N$$, the product $$nr$$ will always be less than some fixed infinite number $$H$$ (in the case of the hyperreals, it can be chosen to be a hyperinteger, i.e., equal to its own integer part/floor). However, to formulate property 2.4.4, one already needs the existence of the infimum, in other words the completeness of the ordered field. But the completeness would imply the Archimedean property automatically.

Therefore I wouldn't worry about their Corollary 2.4.4 if I were you; it is stated as a property of the real numbers, and not formulated as a statement of the Archimedean property. Note that the authors hedge their bets on page 41 by mentioning that it is only "collectively" that the three corollaries can be "referred to as the Archimedean property of $$\mathbb R$$."

Their 2.4.6 simply rules out the existence of an infinite number $$y=H$$ as above (it will never be between two elements of $$\mathbb N$$).

• Okay so 2.4.5 and 2.4.6 are just reinforcing the idea that there are no infinitely big or infinitesimal real numbers. Then can they be used as the definition of the Archimedean Property of R in a proof?
– vina
Commented Nov 2, 2023 at 14:58
• @vina they can be used as the definition of being Archimedean in any ordered field and in particular in R. To understand what is going on it may be helpful to think of some basic examples where the Archimedean property is not satisfied, so as not to feel that you are dealing with a tautology. Commented Nov 2, 2023 at 15:02
• Property 2.4.4 is not "odd" at all, it does not presuppose the existence of "the" infimum (of every nonempty bounded subset) i.e. the completeness of the ordered field. It only tells you that the particular subset $\{1/n\mid n\in\Bbb N\}$ has an infimum and that this infimum is $0.$ You can easily check that this is equivalent to 2.4.5. Commented Nov 2, 2023 at 16:19
• @AnneBauval, the problem is how "infimum" is defined exactly. Before you can check equivalence with other definitions, you need to know what you are talking about :-) Infimum is not a concept in first-order logic as you have to quantify over subsets to define it. My objection stands unless you can clarify how the infimum would be defined. What meaning do you attach to the assertion that the infimum of 1/n over N is $0$? According to your definition of infimum, what would be the infimum of 1/n over N inside the field $\mathbb R^\ast$? Commented Nov 2, 2023 at 16:25
• The general definition of "infimum" (of any subset $S$ of any ordered set) is usual (it may exist or not, but when writing $\inf(S)=0,$ the claim is that it exists and is equal to $0$). Here, the meaning of $\inf(\{1/n\mid n\in\Bbb N\})=0$ is precisely 2.4.5 (given we already know that $\forall n\in\Bbb N,1/n>0$). Commented Nov 2, 2023 at 17:24

This is my final conclusion:

Archimedean Property of an Ordered Field:

"If $$x,y$$ are any positive numbers in the ordered field $$F$$, then there exists $$n$$ in the Natural numbers such that $$nx>y$$"

Archimedean Property of Real Numbers: Version 1

"If $$x,y$$ are any positive real numbers then there exists $$n$$ in the Natural numbers such that $$nx>y$$"

[This is just choosing the ordered field $$F$$ to be the complete ordered field $$\Bbb R$$ of Reals by directly substituting $$\Bbb R$$ into the definition]

Archimedean Property of Real Numbers: Version 2

"If x is any Real number then there exists n in the Natural numbers such that $$n>x$$"

[This definition is derived from the Archimedean Property of an Ordered Field in terms of Real numbers. Specifically, since $$x$$ can be any positive real number, let $$x=1$$ then $$nx>y\implies n>y$$]

Finally, the Archimedean Property of Real Numbers is just saying that there are no "infinitely big" real numbers and the conclusion that Natural Numbers are not bounded above is deduced as a result from the proof of the Archimedean Property.

Does everything seem right?

But I still don't understand what Corollaries 2.4.4 and 2.4.6 mean.

There is a bunch of posts on MSE about the Archimedean property, but let us try to answer your four precise questions, in the context of an arbitrary ordered field $$F$$.

1. $$F$$ has the Archimedean property iff $$\forall x\in F,\exists n\in\Bbb N,x\le n \tag1$$ or equivalently iff $$\forall u,v\in F^+,\exists m\in\Bbb N,mu>v\tag2.$$ and also iff $$\forall x\in F^+,\exists n\in\Bbb N,x\le n \tag3$$ $$(1)\implies(2):$$ given $$u,v>0,$$ $$(1)$$ provides you an $$n\in\Bbb N$$ such that $$v/u\le n,$$ hence $$v/u

$$(2)\implies(3):$$ if $$x>0,$$ $$(2)$$ gives you an $$m\in\Bbb N$$ such that $$m1>x,$$ and a fortiori $$x\le n:=m.$$

$$(3)\implies(1):$$ if $$x\le0$$, any $$n\in\Bbb N$$ will do.

2. Though equivalent to $$(1)$$ i.e. essentially to $$(3)$$, $$(2)$$ is more useful because it looks "more general": roughly (but the proof above treats more precisely the difference -inessential here- between $$\le$$ and $$<$$), $$(3)$$ can be considered as the particular case $$u=1$$ of $$(2).$$

3. 2.4.4 is equivalent to 2.4.5 by a mere application of the definition of $$\inf$$ (details added upon request in edit below). 2.4.5 is equivalent to $$(3)$$ by the change of variable $$t=1/x.$$ 2.4.6, once corrected (replacing $$n-1 < y < n$$ by $$n-1 < y\le n$$), is only seemingly stronger than $$(3)$$: instead of any $$n\in\Bbb N$$ such that $$y\le n,$$ take the smallest one.

4. The Archimedean property for $$\Bbb R$$ is nothing else than all these equivalent formulations, with $$F$$ replaced by $$\Bbb R.$$

Edit 1. Details for the first statement of point 3: 2.4.4 is equivalent to 2.4.5 by a mere application of the definition of $$\inf$$. Indeed, $$\forall n\in\Bbb N,1/n>0$$ holds in every ordered field. Therefore, 2.4.4 and 2.4.5 are both equivalent to $$\forall t>0,\exists n\in\Bbb N,1/n

Edit 2. Equivalence between the new (corrected) version $$(\alpha)$$ of 2.4.6 in the question, and my version $$(\beta)$$, where I previously chose the other way to correct it: $$\begin{cases}(\alpha):&\forall y>0,\exists n\in\Bbb N,n-1\le y0,\exists m\in\Bbb N,m-1 Whenever $$y\in\Bbb N$$ (the only case where there might be some doubt), both properties hold:

• for $$m:=y$$, we have $$m\in\Bbb N$$ and $$m-1 and
• for $$n:=y+1,$$ we have $$n\in\Bbb N$$ and $$n-1\le y
• It is somewhat dubious to claim that "2.4.4 is equivalent to 2.4.5 by a mere application of the definition of inf" as you do. How would you distinguish Archimedean from non-Archimedean fields using Corollary 2.4.4 (used as a definition)? For example, consider the field of Laurent series (or any other non-Archimdean field of your choice). How would you prove using 2.4.4 that this field is non-Archimedean (of course without using the equivalence with another definition)? Commented Nov 5, 2023 at 10:10
• Again: when writing $\inf(S)=0$, the claim is that this $\inf$ exists and is equal to $0$. Every non-Archimedean field fails to satisfy 2.4.4 because for such a field, $\inf(\{1/n: n\in\Bbb N\})$ either does not exist or is $>0$ (actually, it does not exist, but we don't need to prove it). @MikhailKatz Commented Nov 5, 2023 at 14:31
• You are avoiding my question. How do you prove that, other than claiming that "it is easy"? Commented Nov 5, 2023 at 14:31
• You claim that Corollary 2.4.4 gives a suitable definition of an Archimedean field. How would you use such a definition to prove that a given non-Archimedean field is indeed non-Archimedean? Choose your favorite example (a simple example is given by the field of Laurent series). Commented Nov 5, 2023 at 14:37
• Better than an example, again: which step of my general proof (including the recent edit) are you missing? $(3)\iff$2.4.5? 2.4.5$\iff \forall t>0,\exists n\in\Bbb N,1/n<t$? 2.4.4$\iff \forall t>0,\exists n\in\Bbb N,1/n<t$? @MikhailKatz (or don't you accept $(3)$ as a definition of the Archimedean property?) Commented Nov 5, 2023 at 14:39