Math 2.0

Real functions Theorem

Triangle inequality

Builds on Absolute value

Statement

For all real numbers aa and bb, ∣a+b∣≤∣a∣+∣b∣,|a + b| \le |a| + |b|, where ∣⋅∣|\cdot| is the absolute value.

Remarks

Proof: −∣a∣≤a≤∣a∣-|a| \le a \le |a| and −∣b∣≤b≤∣b∣-|b| \le b \le |b|. Add these inequalities to get −(∣a∣+∣b∣)≤a+b≤∣a∣+∣b∣-(|a| + |b|) \le a + b \le |a| + |b|. Equality holds exactly when ab≥0a b \ge 0. For distances, it gives ∣x−z∣≤∣x−y∣+∣y−z∣|x - z| \le |x - y| + |y - z|. The reverse triangle inequality is a consequence.

Reading path

To understand this page from the ground up, read these 13 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 numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe reals form a field, statement, no evidence levelThe reals form a fieldstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe 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…statementAbsolute value, definition, no evidence levelAbsolute valuedefinitionTriangle inequality, statement, no evidence levelTriangle inequalitystatement
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. Ring of integers statement
  7. The reals form a field statement
  8. The rationals form a field statement
  9. The rationals form an ordered field statement
  10. Order on the reals definition
  11. The reals form an ordered field statement
  12. Absolute value definition
  13. Triangle inequality statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangle_inequality, lead section
  2. Web https://en.wikipedia.org/wiki/Absolute_value, Real numbers
About this page
Tag03HF5PMY5T
URIm2://03HF5PMY5T
Kindstatement, theorem
WritingW1 stated
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 03HF5PMY5T.
UsesAbsolute value
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