# 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/