Difference between revisions of "Least upper bound"
m (supremum redirects here, and the two terms are equivalent, so no link out) |
|||
Line 5: | Line 5: | ||
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. | ||
+ | |||
+ | '''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 10:08, 4 November 2006
This article is a stub. Help us out by expanding it.
Given a subset in some larger ordered set
, a least upper bound or supremum, for
is an element
such that
for every
and there is no
with this same property.
If the least upper bound of
is an element of
, it is also the maximum of
. If
, then
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.