Editing Infimum and supremum (section)

=== Least-upper-bound property ===
{{main|Least-upper-bound property}}

The {{em|least-upper-bound property}} is an example of the aforementioned [[Completeness (order theory)|completeness properties]] which is typical for the set of real numbers. This property is sometimes called {{em|Dedekind completeness}}.

If an ordered set <math>S</math> has the property that every nonempty subset of <math>S</math> having an upper bound also has a least upper bound, then <math>S</math> is said to have the least-upper-bound property.  As noted above, the set <math>\R</math> of all real numbers has the least-upper-bound property.  Similarly, the set <math>\Z</math> of integers has the least-upper-bound property; if <math>S</math> is a nonempty subset of <math>\Z</math> and there is some number <math>n</math> such that every element <math>s</math> of <math>S</math> is less than or equal to <math>n,</math> then there is a least upper bound <math>u</math> for <math>S,</math> an integer that is an upper bound for <math>S</math> and is less than or equal to every other upper bound for <math>S.</math> A [[well-order]]ed set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set that {{em|lacks}} the least-upper-bound property is <math>\Q,</math> the set of rational numbers.  Let <math>S</math> be the set of all rational numbers <math>q</math> such that <math>q^2 < 2.</math> Then <math>S</math> has an upper bound (<math>1000,</math> for example, or <math>6</math>) but no least upper bound in <math>\Q</math>: If we suppose <math>p \in \Q</math> is the least upper bound, a contradiction is immediately deduced because between any two reals <math>x</math> and <math>y</math> (including [[square root of 2|<math>\sqrt{2}</math>]] and <math>p</math>) there exists some rational <math>r,</math> which itself would have to be the least upper bound (if <math>p > \sqrt{2}</math>) or a member of <math>S</math> greater than <math>p</math> (if <math>p < \sqrt{2}</math>). Another example is the [[hyperreals]]; there is no least upper bound of the set of positive infinitesimals.

There is a corresponding {{em|greatest-lower-bound property}}; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered set <math>P</math> every bounded subset has a supremum, this applies also, for any set <math>X,</math> in the function space containing all functions from <math>X</math> to <math>P,</math> where <math>f \leq g</math> if and only if <math>f(x) \leq g(x)</math> for all <math>x \in X.</math> For example, it applies for real functions, and, since these can be considered special cases of functions, for real <math>n</math>-tuples and sequences of real numbers.

The [[least-upper-bound property]] is an indicator of the suprema.