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

From Wikiversity
Jump to navigation Jump to search


Definition  

A field is called an ordered field if there is a relation (larger than) between the elements of fulfilling the following properties ( means or )

  1. For two elements we have either or or .
  2. From and one may deduce (for any ).
  3. implies (for any ).
  4. From and one may deduce (for any ).


Lemma

In an ordered field the following properties hold.

  1. .
  2. holds if and only if holds.
  3. holds if and only if holds.
  4. holds if and only if holds.
  5. and imply .
  6. and imply .
  7. and imply .
  8. and imply .
  9. and imply .
  10. and imply .

Proof

See Exercise.



Lemma

In an ordered field the following properties holds.

  1. From one can deduce .
  2. From one can deduce .
  3. For we have if and only if .
  4. From one can deduce .
  5. For positive elements the relation is equivalent with .

Proof

See Exercise.



Definition  

Let be an ordered field. is called Archimedean, if the following Archimedean axiom holds, i.e. if for every there exists a natural number such that


Lemma

  1. For with there exists such that .
  2. For there exists a natural number such that .
  3. For two real numbers there exists a rational number (with , ) such that

Proof  

(1). We consider . Because of the Archimedean axiom there exists some natural number with . Since is positive, due to fact also holds. For (2) and (3) see exercise.



Definition  

For real numbers , , we call

  1. the closed interval.
  2. the open interval.
  3. the half-open interval (closed on the right).
  4. the half-open interval (closed on the left).