Editing Infimum and supremum (section)

=== Suprema ===
* The supremum of the set of numbers <math>\{1, 2, 3\}</math> is <math>3.</math> The number <math>4</math> is an upper bound, but it is not the least upper bound, and hence is not the supremum.
* <math>\sup \{ x \in \R : 0 < x < 1\} = \sup \{ x \in \R : 0 \leq x \leq 1\} = 1.</math>
* <math>\sup \left\{ (-1)^n - \tfrac{1}{n} : n = 1, 2, 3, \ldots \right\} = 1.</math>
* <math>\sup \{ a + b : a \in A, b \in B \} = \sup A + \sup B.</math>
* <math>\sup \left\{ x \in \Q : x^2 < 2 \right\} = \sqrt{2}.</math>

In the last example, the supremum of a set of [[Rational number|rationals]] is [[Irrational number|irrational]], which means that the rationals are [[Complete space|incomplete]].

One basic property of the supremum is
<math display=block>\sup \{ f(t) + g(t) : t \in A \} ~\leq~ \sup \{ f(t) : t \in A \} + \sup \{ g(t) : t \in A \}</math>
for any [[Functional (mathematics)|functionals]] <math>f</math> and <math>g.</math>

The supremum of a subset <math>S</math> of <math>(\N, \mid\,)</math> where <math>\,\mid\,</math> denotes "[[Divisor|divides]]", is the [[lowest common multiple]] of the elements of <math>S.</math>

The supremum of a set <math>S</math> containing subsets of some set <math>X</math> is the [[Union (set theory)|union]] of the subsets when considering the partially ordered set <math>(P(X), \subseteq)</math>, where <math>P</math> is the [[power set]] of <math>X</math> and <math>\,\subseteq\,</math> is [[subset]].