Archimedean Property

The Archimedean Property is a fundamental consequence of the completeness axiom. It asserts that there are no "infinitely large" or "infinitely small" positive real numbers: given any two positive reals, a sufficiently large multiple of the smaller one will exceed the larger one. This property is essential for density arguments, decimal expansions, and many approximation results in analysis.

Statement

Theorem1.1Archimedean Property

For all $x, y \in \mathbb{R}$ with $x > 0$ , there exists $n \in \mathbb{N}$ such that

$nx > y.$

RemarkEquivalent formulation

Equivalently: for every $\epsilon > 0$ , there exists $n \in \mathbb{N}$ such that $1/n < \epsilon$ . This formulation says that the sequence $\{1/n\}$ converges to zero — there are no "infinitesimal" positive reals that are smaller than $1/n$ for all $n$ .

Proof

Suppose, for the sake of contradiction, that $nx \leq y$ for all $n \in \mathbb{N}$ . Then the set

$S = \{nx \mid n \in \mathbb{N}\}$

is bounded above by $y$ . By the completeness axiom (least upper bound property), $S$ has a supremum in $\mathbb{R}$ . Let $\alpha = \sup S$ .

Since $(n+1)x \in S$ for all $n \in \mathbb{N}$ , we have

$(n+1)x \leq \alpha \quad \text{for all } n \in \mathbb{N}.$

Rearranging, $nx \leq \alpha - x$ for all $n \in \mathbb{N}$ . Thus $\alpha - x$ is an upper bound for $S$ . But $x > 0$ , so $\alpha - x < \alpha$ , contradicting the fact that $\alpha$ is the least upper bound of $S$ .

Therefore, our assumption was wrong, and there must exist $n \in \mathbb{N}$ with $nx > y$ .

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RemarkDependence on completeness

The proof relies crucially on the completeness axiom. Over the rationals $\mathbb{Q}$ (which is not complete), the Archimedean property still holds, but the proof uses the well-ordering of $\mathbb{N}$ instead. The point here is that completeness of $\mathbb{R}$ implies the Archimedean property — this is not true for all ordered fields.

Corollaries and applications

Theorem1.2No infinitesimals in R

There is no positive real number $\epsilon > 0$ such that $\epsilon < 1/n$ for all $n \in \mathbb{N}$ .

Proof

If such an $\epsilon$ existed, then by the Archimedean property applied to $\epsilon$ and $1$ , there would exist $n \in \mathbb{N}$ with $n\epsilon > 1$ , i.e., $\epsilon > 1/n$ , contradiction.

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Theorem1.3Integers are unbounded

The set $\mathbb{Z}$ of integers is unbounded in $\mathbb{R}$ : for every $M \in \mathbb{R}$ , there exists $n \in \mathbb{Z}$ with $n > M$ .

Proof

Given $M \in \mathbb{R}$ , apply the Archimedean property with $x = 1$ and $y = M$ to find $n \in \mathbb{N}$ with $n > M$ .

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Theorem1.4Sequence 1/n converges to 0

$\lim_{n \to \infty} \frac{1}{n} = 0.$

Proof

Let $\epsilon > 0$ . By the Archimedean property, there exists $N \in \mathbb{N}$ such that $1/N < \epsilon$ . Then for all $n \geq N$ , we have

$\frac{1}{n} \leq \frac{1}{N} < \epsilon.$

Thus $|1/n - 0| < \epsilon$ for all $n \geq N$ , so $1/n \to 0$ .

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ExampleDecimal expansions

The Archimedean property underlies decimal (or binary, etc.) expansions of real numbers. Given $x \in [0, 1)$ , we can find $d_1 \in \{0, 1, \ldots, 9\}$ such that $d_1/10 \leq x < (d_1+1)/10$ (by repeated application of the Archimedean property to subdivide $[0,1]$ into tenths). Continuing recursively, we obtain $x = 0.d_1 d_2 d_3 \ldots$

ExampleApproximation by rationals

For any $x \in \mathbb{R}$ and $\epsilon > 0$ , there exists $q \in \mathbb{Q}$ with $|x - q| < \epsilon$ . This follows from the density of $\mathbb{Q}$ in $\mathbb{R}$ , which in turn relies on the Archimedean property (see Density of Rationals).

ExampleApproximating √2

To approximate $\sqrt{2}$ to within $10^{-6}$ , we use the Archimedean property to find $n$ large enough that $1/n < 10^{-6}$ , then subdivide $[1, 2]$ into intervals of width $1/n$ to locate $\sqrt{2}$ within one such interval. This is the basis for numerical algorithms.

Non-Archimedean fields

RemarkNon-Archimedean ordered fields exist

There exist ordered fields that do not satisfy the Archimedean property. For example, the field $\mathbb{R}(t)$ of rational functions in an indeterminate $t$ , ordered so that $t$ is "infinitely large" (i.e., $t > n$ for all $n \in \mathbb{N}$ ), is a non-Archimedean ordered field. In this field, no multiple of $1$ exceeds $t$ .

Such fields arise in model theory (nonstandard analysis) and algebraic geometry (valuations on function fields). But $\mathbb{R}$ itself is Archimedean, thanks to completeness.

ExampleHyperreal numbers

The hyperreal numbers ${}^*\mathbb{R}$ (used in nonstandard analysis) extend $\mathbb{R}$ by adding infinitesimals and infinitely large elements. In ${}^*\mathbb{R}$ , there exist $\epsilon > 0$ with $\epsilon < 1/n$ for all standard $n \in \mathbb{N}$ . The hyperreals are not Archimedean, and completeness (in the sense of least upper bounds) fails.

Historical context

RemarkEudoxus and Archimedes

The Archimedean property is named after Archimedes of Syracuse (c. 287–212 BCE), though the idea appears earlier in Eudoxus's theory of proportions (c. 370 BCE). Euclid's Elements (Book V, Definition 4) includes a version of the Archimedean property as an axiom for magnitudes.

Archimedes used the principle to compute areas and volumes via the method of exhaustion: given two magnitudes $A$ and $B$ , one can find a multiple of the smaller that exceeds the larger, allowing approximation to arbitrary precision.

RemarkModern formulation

The modern formulation of the Archimedean property as a consequence of completeness (rather than an axiom) is due to the 19th-century rigorization of analysis by Cauchy, Weierstrass, and Dedekind. The proof given here, via the least upper bound property, is standard in modern real analysis textbooks.

Summary

The Archimedean property is a cornerstone of real analysis:

It states that $\mathbb{R}$ has no infinitesimals or infinitely large elements.
It is a consequence of the completeness axiom.
It implies that $\{1/n\} \to 0$ , that $\mathbb{N}$ is unbounded, and that $\mathbb{Q}$ is dense in $\mathbb{R}$ .
It underpins approximation arguments, decimal expansions, and the intermediate value theorem.

See Density of Rationals for an immediate application, and Proof of Archimedean Property for an alternative proof via the well-ordering principle.