Math 2.0

Real functions Proof

Proof of the approximation property of the supremum

Builds on SupremumSet bounded aboveThe reals form an ordered field

Proof

We argue by contradiction.

  1. Let SS be a nonempty set of reals that is bounded above, let b=sup⁡Sb = \sup S be its supremum, and let c<bc < b.
  2. Suppose that no s∈Ss \in S satisfies c<s≤bc < s \le b.
  3. Let s∈Ss \in S. Since bb is an upper bound of SS, s≤bs \le b. By step 2, c<sc < s fails.
  4. The order of ℝ\R is total, by the ordered field property. So s≤cs \le c.
  5. Thus cc is an upper bound of SS. Since bb is the least upper bound, b≤cb \le c.
  6. This contradicts c<bc < b. So some s∈Ss \in S satisfies c<s≤bc < s \le b.

Idea

A number below the supremum cannot be an upper bound, because the supremum is the least one. So some element of the set lies above it.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first. The path takes the proof Proof that the reals form an ordered field, with the Cauchy sequence construction.

Show the path as a graph
Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementBinary relation, definition, no evidence levelBinary relationdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionProof that the reals form an ordered field, with the Cauchy sequence construction, proof, no evidence levelProof that the reals …proofThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statementProof of the approximation property of the supremum, proof, no evidence levelProof of the approxim…proof
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. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Rational numbers object
  7. Real numbers object
  8. Set bounded above definition
  9. Set definition
  10. Ring of integers statement
  11. Implication definition
  12. Predicate definition
  13. The reals form a field statement
  14. Supremum definition
  15. Ordered pair definition
  16. The rationals form a field statement
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. The rationals form an ordered field statement
  21. Binary relation definition
  22. Order on the reals definition
  23. Antisymmetric relation definition
  24. Reflexive relation definition
  25. Transitive relation definition
  26. Preorder definition
  27. Partial order definition
  28. Total order definition
  29. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  30. The reals form an ordered field statement
  31. Proof of the approximation property of the supremum proof

Where this leads

1651 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Identifying infima and suprema
About this page
Tag2WFSYYVW5H
URIm2://2WFSYYVW5H
Kindproof
WritingW0 stub
MSC26A03
ProvesApproximation property of the supremum
UsesSupremum, Set bounded above, The reals form an ordered field
MethodsProof by contradiction
Changed4 November 2026

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