# Archimedean Property

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Let \( ( S, \circ \) be a closed algebraic structure with a Norm (e.g. real numbers with absolute value as a norm). Then, the norm \( n\) satisfies the Archimedean property on \(S\) if and only if

\[ \forall a, b \in S, n(a) < n(b) \Rightarrow \exists m \in N \text{ such that } n ( m \cdot a) > n (b) \]

Corollary: For all real numbers \( r \), there exists an integer \(n\) such that \( n > r \).

Note: The field of rational functions of \(x\) does not satisfy the Archimedean property.

#### Contents

- Forms of Completeness
- Least Upper Bound Property
- Cauchy Completeness
- Nested Intervals Theorem
- Monotone Convergence Theorem
- Bolzano Weierstrass Theorem
- Archimedean Property
- Example 2
- Example 3
- Example 4
- Example 5
- Example 6
- Example 7
- Example 8
- Example 9
- Example 10
- Example 11
- Example 12
- Example 13
- Example 14
- Example 15
- Example 16
- Example 17
- Example 18
- Example 19
- Example 20

## Forms of Completeness

Frequently, Dedicand cuts are used axiomatically to construct the Reals. The Reals definitely have Dedicand completeness. :) The rationals do not.

A la wikipedia:

## Least Upper Bound Property

Real number system has the property that every non-empty subset of R which is bounded above has a least upper bound. This property is called Least Upper Bound Property.

## Cauchy Completeness

Cauchy completeness is the statement that every Cauchy sequence of real numbers converges. Cauchy Sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Mathematically, for a given \(\epsilon>0\), there exists a natural number \(N\) such that \(|x_m-x_n|<\epsilon\ \forall\ m,n>N\).

## Nested Intervals Theorem

For each \(n\), let \(I_n = [a_n, b_n]\) be a (non-empty) bounded interval of real numbers such that \[I_1 \supset\ I_2 \supset I_3 \supset \cdots \supset I_n \supset I_{n+1} \supset \cdots\] and \(\displaystyle\lim_{n\rightarrow 0} (b_n-a_n)=0\). Then \(\displaystyle\bigcap_{n=1}^{\infty} I_n\) contains only one point.

The nested interval theorem states that the intersection of all of the intervals \(I_n\) is nonempty.

Source: Springer's "Real Analysis and Applications"

## Monotone Convergence Theorem

The monotone convergence theorem states that every nondecreasing, bounded sequence of real numbers converges.

## Bolzano Weierstrass Theorem

The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.

## Archimedean Property

Prove that Real numbers follow Archimedean Property

Source: Springer's "Real Analysis and Applications"

## Example 2

Prove that Hyperreal Numbers do not follow Archimedian Property

## Example 3

Existence of rational/irrational number between two real numbers

## Example 4

The uncountability of the reals.

## Example 5

Arithmetic Properties

## Example 6

Properties of Inequalities in the Reals

## Example 7

Let \(x\in R\). Show that there exists an integer \(m\) such that \(m\le x < m+1\) and an integer \(l\) such that \(x<l\le x+1\).

## Example 8

Suppose that \(\alpha,\beta\) are two real numbers satisfying \(\alpha<\beta\). Show that there exist \(n_1,n_2 \in N\) such that \(\alpha<\alpha+\frac{1}{n_1}<\beta\) and \(\alpha<\beta-\frac{1}{n_2}<\beta\)

## Example 9

....

## Example 10

## Example 11

## Example 12

## Example 13

## Example 14

## Example 15

## Example 16

## Example 17

## Example 18

## Example 19

## Example 20

**Cite as:**Archimedean Property.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/archimedean-property/