# Real numbers/Ordering axioms/Archimedes/Intervals/Introduction/2/Section

## Definition

A field ${\displaystyle {}K}$ is called an ordered field if there is a relation ${\displaystyle {}>}$ (larger than) between the elements of ${\displaystyle {}K}$ fulfilling the following properties (${\displaystyle {}a\geq b}$ means ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$)

1. For two elements ${\displaystyle {}a,b\in K}$ we have either ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$ or ${\displaystyle {}b>a}$.
2. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}b\geq c}$ one may deduce ${\displaystyle {}a\geq c}$ (for any ${\displaystyle {}a,b,c\in K}$).
3. ${\displaystyle {}a\geq b}$ implies ${\displaystyle {}a+c\geq b+c}$ (for any ${\displaystyle {}a,b,c\in K}$).
4. From ${\displaystyle {}a\geq 0}$ and ${\displaystyle {}b\geq 0}$ one may deduce ${\displaystyle {}ab\geq 0}$ (for any ${\displaystyle {}a,b\in K}$).

## Lemma

In an ordered field the following properties hold.

1. ${\displaystyle {}1\geq 0}$.
2. ${\displaystyle {}a\geq 0}$ holds if and only if ${\displaystyle {}-a\leq 0}$ holds.
3. ${\displaystyle {}a\geq b}$ holds if and only if ${\displaystyle {}a-b\geq 0}$ holds.
4. ${\displaystyle {}a\geq b}$ holds if and only if ${\displaystyle {}-a\leq -b}$ holds.
5. ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\geq d}$ imply ${\displaystyle {}a+c\geq b+d}$.
6. ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\geq 0}$ imply ${\displaystyle {}ac\geq bc}$.
7. ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\leq 0}$ imply ${\displaystyle {}ac\leq bc}$.
8. ${\displaystyle {}a\geq b\geq 0}$ and ${\displaystyle {}c\geq d\geq 0}$ imply ${\displaystyle {}ac\geq bd}$.
9. ${\displaystyle {}a\geq 0}$ and ${\displaystyle {}b\leq 0}$ imply ${\displaystyle {}ab\leq 0}$.
10. ${\displaystyle {}a\leq 0}$ and ${\displaystyle {}b\leq 0}$ imply ${\displaystyle {}ab\geq 0}$.

### Proof

See Exercise.
${\displaystyle \Box }$

## Lemma

In an ordered field the following properties holds.

1. From ${\displaystyle {}x>0}$ one can deduce ${\displaystyle {}x^{-1}>0}$.
2. From ${\displaystyle {}x<0}$ one can deduce ${\displaystyle {}x^{-1}<0}$.
3. For ${\displaystyle {}x>0}$ we have ${\displaystyle {}x\geq 1}$ if and only if ${\displaystyle {}x^{-1}\leq 1}$.
4. From ${\displaystyle {}x\geq y>0}$ one can deduce ${\displaystyle {}x^{-1}\leq y^{-1}}$.
5. For positive elements ${\displaystyle {}x,y}$ the relation ${\displaystyle {}x\geq y}$ is equivalent with ${\displaystyle {}{\frac {x}{y}}\geq 1}$.

### Proof

See Exercise.
${\displaystyle \Box }$

## Definition

Let ${\displaystyle {}K}$ be an ordered field. ${\displaystyle {}K}$ is called Archimedean, if the following Archimedean axiom holds, i.e. if for every ${\displaystyle {}x\in K}$ there exists a natural number ${\displaystyle {}n}$ such that

${\displaystyle {}n\geq x\,.}$

## Lemma

1. For ${\displaystyle {}x,y\in \mathbb {R} }$ with ${\displaystyle {}x>0}$ there exists ${\displaystyle {}n\in \mathbb {N} }$ such that ${\displaystyle {}nx\geq y}$.
2. For ${\displaystyle {}x>0}$ there exists a natural number ${\displaystyle {}n}$ such that ${\displaystyle {}{\frac {1}{n}}.
3. For two real numbers ${\displaystyle {}x there exists a rational number ${\displaystyle {}n/k}$ (with ${\displaystyle {}n\in \mathbb {Z} }$, ${\displaystyle {}k\in \mathbb {N} _{+}}$) such that
${\displaystyle {}x<{\frac {n}{k}}

### Proof

(1). We consider ${\displaystyle {}y/x}$. Because of the Archimedean axiom there exists some natural number ${\displaystyle {}n}$ with ${\displaystyle {}n\geq y/x}$. Since ${\displaystyle {}x}$ is positive, due to fact also ${\displaystyle {}nx\geq y}$ holds. For (2) and (3) see exercise.

${\displaystyle \Box }$

## Definition

For real numbers ${\displaystyle {}a,b}$, ${\displaystyle {}a\leq b}$, we call

1. ${\displaystyle {}[a,b]={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x\leq b\right\}}}$ the closed interval.
2. ${\displaystyle {}]a,b[={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x the open interval.
3. ${\displaystyle {}]a,b]={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x\leq b\right\}}}$ the half-open interval (closed on the right).
4. ${\displaystyle {}[a,b[={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x the half-open interval (closed on the left).