Math 2.0

Real functions Proof

Proof of the existence of the infimum by negation

Builds on Completeness axiomInfimumSupremumSet bounded aboveThe reals form an ordered field

Proof

  1. Let S⊆ℝS \subseteq \R be nonempty and bounded below. Let mm be a lower bound of SS.
  2. In the ordered field ℝ\R, u≤vu \le v implies −v≤−u-v \le -u. Indeed, add −u−v-u - v to both sides.
  3. Let −S={−s:s∈S}-S = \set{-s : s \in S}. It is nonempty, because SS is nonempty.
  4. For each s∈Ss \in S we have m≤sm \le s, so −s≤−m-s \le -m by step 2. Thus −m-m is an upper bound of −S-S.
  5. By the completeness axiom, −S-S has a supremum bb. Let a=−ba = -b.
  6. The number aa is a lower bound of SS. For s∈Ss \in S, we have −s∈−S-s \in -S, so −s≤b-s \le b. By step 2, a=−b≤sa = -b \le s.
  7. Let ℓ\ell be any lower bound of SS. As in step 4, −ℓ-\ell is an upper bound of −S-S.
  8. Since bb is the least upper bound, b≤−ℓb \le -\ell. By step 2, ℓ≤−b=a\ell \le -b = a.
  9. By steps 6 and 8, aa is a lower bound of SS that is larger than or equal to each lower bound. So aa is an infimum of SS.

Idea

Negation reverses the order, so it turns lower bounds into upper bounds. The supremum of the negated set exists by completeness, and its negative is the infimum.

Reading path

To understand this page from the ground up, read these 33 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.

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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 fieldstatementInfimum, definition, no evidence levelInfimumdefinitionSupremum, 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 quantifierdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementCartesian 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 existence of the infimum by negation, proof, no evidence levelProof of the existenc…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. Infimum definition
  15. Supremum definition
  16. Ordered pair definition
  17. The rationals form a field statement
  18. Universal quantifier definition
  19. Completeness axiom statement
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Binary relation definition
  24. Order on the reals definition
  25. Antisymmetric relation definition
  26. Reflexive relation definition
  27. Transitive relation definition
  28. Preorder definition
  29. Partial order definition
  30. Total order definition
  31. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  32. The reals form an ordered field statement
  33. Proof of the existence of the infimum by negation proof

Where this leads

880 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Duality
About this page
TagK5787SF394
URIm2://K5787SF394
Kindproof
WritingW0 stub
MSC26A03
ProvesExistence of the infimum
UsesCompleteness axiom, Infimum, Supremum, Set bounded above, The reals form an ordered field
Changed4 November 2026

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