Archimedean Property of Real Numbers: Statement, Proof, Example

The Archimedean property of real numbers states that for positive real numbers x and y, there is an integer n>0 such that nx>y. This principle is named after the ancient Greek mathematician Archimedes. In this post, we will learn about the Archimedean property of real numbers along with its proof and applications.

Statement of Archimedean Property of Real Numbers

Let x and y be two real numbers with x>0. Then there exists a positive integer n such that nx>y.

Proof of Archimedean Property of Real Numbers

We will prove the Archimedean property using the completeness axiom of real numbers. Consider the set

S = {nx : n ∈ ℕ}.

For a contradiction, assume that nx ≤ y for all n ∈ ℕ. This makes y is an upper bound of S. As S is non-empty and bounded above, by the completeness axiom of real numbers Sup(S) exists.

Let M = Sup(S).

As x>0 given, M-x cannot be an upper bound of S.

So M-x < mx for some mx ∈ S.

⇒ M < (m+1)x

As (m+1)x ∈ S, we conclude that M = Sup(S) is not true, which is a contradiction. Thus, our assumption nx ≤ y for all n ∈ ℕ is wrong.

In other words, nx>y for some positive integer n. This completes the proof of the Archimedean property of ℝ. ♣

Related Topics:

• Completeness Axiom of Real Numbers
• Supremum & Infimum of a Set

Archimedean Property of Rational Numbers

Question: The set Q of rational numbers satisfies Archimedean property.

Solution:

To prove the set Q of rational numbers satisfies Archimedean property, we need to show that if x is a positive rational and y is any rational number, then there exists n ∈ ℕ such that

nx > y.

Let us prove the case where y>0 and x<y. For a contradiction, assume that nx ≤ y for all n ∈ ℕ.

⇒ $nx \times \dfrac{1}{y}$ ≤ 1 < m where m ∈ ℕ and m>1.

⇒ $\dfrac{n}{m} \leq \dfrac{y}{x} \cdot \dfrac{n}{m}$ is any rational number. Note that for a fixed x and y, the number y/x is a fixed rational number. So this result is not true always.

That is, nx ≤ y is not true always.

In other words, we have some integer n ∈ ℕ such that nx > y. This proves that the set of rational numbers satisfies the Archimedean property of real numbers.

Example

For every positive real number x, ∃ n ∈ ℕ such that n-1 ≤ x < n.

Proof:

As the set ℕ of natural numbers is unbounded above, ∃ n ∈ ℕ such that x<n. So it remains to show that n-1 ≤ x.

Let S = {m>x : m ∈ ℕ}.

Note that S is non-empty. Because, n ∈ S and n is the least element of S.

So n-1 ∉ S.

⇒ n-1 ≤ x.

Thus, we have shown that n-1 ≤ x < n for some natural number n.

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