Math 2.0

Real functions Proof

Proof of the triangle inequality for real numbers

Builds on Absolute valueThe reals form an ordered field

Proof

  1. Let tt be real. We show −∣t∣≤t≤∣t∣-\abs{t} \le t \le \abs{t}, with the absolute value.
  2. If t≥0t \ge 0, then ∣t∣=t\abs{t} = t and −∣t∣=−t≤0≤t-\abs{t} = -t \le 0 \le t. If t<0t < 0, then ∣t∣=−t>0>t\abs{t} = -t > 0 > t and −∣t∣=t-\abs{t} = t.
  3. Let r≥0r \ge 0 and let ss be real with −r≤s≤r-r \le s \le r. Then ∣s∣≤r\abs{s} \le r. If s≥0s \ge 0, then ∣s∣=s≤r\abs{s} = s \le r. If s<0s < 0, then ∣s∣=−s≤r\abs{s} = -s \le r.
  4. In the ordered field ℝ\R, if u≤vu \le v and u′≤v′u' \le v', then u+u′≤v+u′≤v+v′u + u' \le v + u' \le v + v', by adding u′u' and then vv.
  5. Let aa and bb be real. By step 2, −∣a∣≤a≤∣a∣-\abs{a} \le a \le \abs{a} and −∣b∣≤b≤∣b∣-\abs{b} \le b \le \abs{b}.
  6. By step 4, −(∣a∣+∣b∣)≤a+b≤∣a∣+∣b∣-(\abs{a} + \abs{b}) \le a + b \le \abs{a} + \abs{b}.
  7. Here r=∣a∣+∣b∣≥0r = \abs{a} + \abs{b} \ge 0. By step 3 with s=a+bs = a + b, ∣a+b∣≤∣a∣+∣b∣\abs{a + b} \le \abs{a} + \abs{b}.

Idea

Each number lies between minus its absolute value and its absolute value. Adding two such bounds traps a+ba + b between −(∣a∣+∣b∣)-(\abs{a} + \abs{b}) and ∣a∣+∣b∣\abs{a} + \abs{b}.

Reading path

To understand this page from the ground up, read these 30 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, 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 fieldstatementOrdered 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…statementAbsolute value, definition, no evidence levelAbsolute valuedefinitionProof of the triangle inequality for real numbers, proof, no evidence levelProof of the triangle…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 definition
  9. Ring of integers statement
  10. Implication definition
  11. Predicate definition
  12. The reals form a field statement
  13. Ordered pair definition
  14. The rationals form a field statement
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Binary relation definition
  20. Order on the reals definition
  21. Antisymmetric relation definition
  22. Reflexive relation definition
  23. Transitive relation definition
  24. Preorder definition
  25. Partial order definition
  26. Total order definition
  27. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  28. The reals form an ordered field statement
  29. Absolute value definition
  30. Proof of the triangle inequality for real numbers proof

Where this leads

1457 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Triangle_Inequality/Real_Numbers
About this page
TagHRTKMX90SJ
URIm2://HRTKMX90SJ
Kindproof
WritingW0 stub
MSC26A03
ProvesTriangle inequality
UsesAbsolute value, The reals form an ordered field
Changed4 November 2026

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