Math 2.0

statement corollary

Reverse 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}.

Explanations

L1 human

Two numbers that are close to each other have absolute values that are also close. The difference of the distances to 00 is at most the distance between the two numbers.

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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
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Triangle_inequality, Reverse triangle inequality

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