Math 2.0

Real functions Proposition

Approximation property of the supremum

Builds on SupremumCompleteness axiom

Statement

Let SS be a nonempty subset of the real numbers that is bounded above, and let b=sup⁡Sb = \sup S be its supremum. For each real number c<bc < b, there is an element ss of SS with c<s≤bc < s \le b.

Remarks

Proof: if no such ss exists, then cc is an upper bound of SS that is less than bb. This contradicts the definition of the supremum. An equivalent form: for each ε>0\varepsilon > 0, there is ss in SS with b−ε<sb - \varepsilon < s. For an infimum a=inf⁡Sa = \inf S and each c>ac > a, there is ss in SS with a≤s<ca \le s < c.

Reading path

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Real numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementApproximation property of the supremum, statement, no evidence levelApproximation propert…statement
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  1. Real numbers object
  2. Set bounded above definition
  3. Supremum definition
  4. Completeness axiom statement
  5. Approximation property of the supremum statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Identifying infima and suprema
About this page
Tag0M8TGGP9JP
URIm2://0M8TGGP9JP
Kindstatement, proposition
WritingW0 stub
MSC26A03
StatusThis copy of the site has no build data. Run m2 status 0M8TGGP9JP.
UsesSupremum, Completeness axiom

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