Difference between revisions of "Least upper bound"

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If the least upper bound <math>M</math> of <math>S</math> is an element of <math>S</math>, it is also the [[maximum]] of <math>S</math>.  If <math>M \not\in S</math>, then <math>S</math> has no maximum.
 
If the least upper bound <math>M</math> of <math>S</math> is an element of <math>S</math>, it is also the [[maximum]] of <math>S</math>.  If <math>M \not\in S</math>, then <math>S</math> has no maximum.
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'''The Least Upper Bound Axiom''': This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear, but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.
  
  
 
[[Category:Definition]]
 
[[Category:Definition]]

Revision as of 09:08, 4 November 2006

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Given a subset $S$ in some larger ordered set $R$, a least upper bound or supremum, for $S$ is an element $\displaystyle M \in R$ such that $s \leq M$ for every $s \in S$ and there is no $m < M$ with this same property.

If the least upper bound $M$ of $S$ is an element of $S$, it is also the maximum of $S$. If $M \not\in S$, then $S$ has no maximum.

The Least Upper Bound Axiom: This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear, but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.