Math 2.0

Real functions Theorem

The reals are Archimedean

Builds on The reals form an ordered fieldArchimedean ordered fieldCompleteness axiom

Statement

The ordered field ℝ\mathbb{R} of real numbers is an Archimedean ordered field. That is, for all real numbers xx and yy with y>0y > 0, there is a natural number nn with ny>xn y > x.

Remarks

The proof uses the completeness axiom. If ℕ\mathbb{N} were bounded above, let b=sup⁡ℕb = \sup \mathbb{N}. Then b−1b - 1 is not an upper bound, so n>b−1n > b - 1 for some nn, and n+1>bn + 1 > b, a contradiction. Thus each complete ordered field is Archimedean.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionRing of integers, statement, no evidence levelRing of integersstatementArchimedean ordered field, definition, no evidence levelArchimedean ordered f…definitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinitionThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statementThe reals are Archimedean, statement, no evidence levelThe reals are Archime…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Ordered field definition
  4. Rational numbers object
  5. Real numbers object
  6. Set bounded above definition
  7. Ring of integers statement
  8. Archimedean ordered field definition
  9. The reals form a field statement
  10. Supremum definition
  11. The rationals form a field statement
  12. Completeness axiom statement
  13. The rationals form an ordered field statement
  14. Order on the reals definition
  15. The reals form an ordered field statement
  16. The reals are Archimedean statement

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Archimedean_property, Archimedean property of the real numbers
About this page
Tag0A3GWZ86AY
URIm2://0A3GWZ86AY
Kindstatement, theorem
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 0A3GWZ86AY.
UsesThe reals form an ordered field, Archimedean ordered field, Completeness axiom

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