Math 2.0

Real functions Corollary

Reverse triangle inequality

Builds on Triangle inequality

Statement

For all real numbers aa and bb, ∣∣a∣−∣b∣∣≤∣a−b∣.\bigl| |a| - |b| \bigr| \le |a - b|.

Remarks

Proof: the triangle inequality gives ∣a∣=∣(a−b)+b∣≤∣a−b∣+∣b∣|a| = |(a - b) + b| \le |a - b| + |b|, thus ∣a∣−∣b∣≤∣a−b∣|a| - |b| \le |a - b|. Exchange aa and bb for the other sign. It shows that the function x↦∣x∣x \mapsto |x| is continuous on ℝ\mathbb{R}.

Reading path

To understand this page from the ground up, read these 14 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 inequalitystatementReverse triangle inequality, statement, no evidence levelReverse triangle ineq…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. 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
  14. Reverse triangle inequality statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangle_inequality, Reverse triangle inequality
About this page
Tag0C9S13WKMF
URIm2://0C9S13WKMF
Kindstatement, corollary
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 0C9S13WKMF.
UsesTriangle inequality

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