Editing Infimum and supremum (section)

===Properties===

If <math>A</math> is any set of real numbers then <math>A \neq \varnothing</math> if and only if <math>\sup A \geq \inf A,</math> and otherwise <math>-\infty = \sup \varnothing < \inf \varnothing = \infty.</math>{{sfn|Rockafellar|Wets|2009|pp=1-2}} 

'''Set inclusion'''

If <math>A \subseteq B</math> are sets of real numbers then <math>\inf A \geq \inf B</math> (if <math>A = \varnothing</math> this reads as <math>\inf B \le \infty</math>) and <math>\sup A \leq \sup B.</math>

'''Image under functions'''
If <math>f \colon \mathbb{R} \to \mathbb{R}</math> is a nonincreasing function, then 
<math>f (\inf(S)) \le \inf (f (S))</math> and <math>\sup(f(S))</math>, where the image is defined as <math>f(S) \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \{f(s) : s \in S\}.</math>

'''Identifying infima and suprema'''

If the infimum of <math>A</math> exists (that is, <math>\inf A</math> is a real number) and if <math>p</math> is any real number then <math>p = \inf A</math> if and only if <math>p</math> is a lower bound and for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon < p + \epsilon.</math> 
Similarly, if <math>\sup A</math> is a real number and if <math>p</math> is any real number then <math>p = \sup A</math> if and only if <math>p</math> is an upper bound and if for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon > p - \epsilon.</math>

'''Relation to limits of sequences'''

If <math>S \neq \varnothing</math> is any non-empty set of real numbers then there always exists a non-decreasing sequence <math>s_1 \leq s_2 \leq \cdots</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \sup S.</math> Similarly, there will exist a (possibly different) non-increasing sequence <math>s_1 \geq s_2 \geq \cdots</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \inf S.</math>
In particular, the infimum and supremum of a set belong to its [[Closure (topology)|closure]] if <math>\inf S \in \mathbb{R}</math> then <math>\inf S \in \bar{S}</math> and if  <math>\sup S \in \mathbb{R}</math> then <math>\sup S \in \bar{S}</math>

Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from [[topology]] that if <math>f</math> is a [[Continuous function (topology)|continuous function]] and <math>s_1, s_2, \ldots</math> is a sequence of points in its domain that converges to a point <math>p,</math> then <math>f\left(s_1\right), f\left(s_2\right), \ldots</math> necessarily converges to <math>f(p).</math> 
It implies that if <math>\lim_{n \to \infty} s_n = \sup S</math> is a real number (where all <math>s_1, s_2, \ldots</math> are in <math>S</math>) and if <math>f</math> is a continuous function whose domain contains <math>S</math> and <math>\sup S,</math> then 
<math display=block>f(\sup S) = f\left(\lim_{n \to \infty} s_n\right) = \lim_{n \to \infty} f\left(s_n\right),</math>
which (for instance) guarantees<ref group=note>Since <math>f\left(s_1\right), f\left(s_2\right), \ldots</math> is a sequence in <math>f(S)</math> that converges to <math>f(\sup S),</math> this guarantees that <math>f(\sup S)</math> belongs to the [[Closure (topology)|closure]] of <math>f(S).</math></ref> that <math>f(\sup S)</math> is an [[adherent point]] of the set <math>f(S) \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \{f(s) : s \in S\}.</math>
If in addition to what has been assumed, the continuous function <math>f</math> is also an increasing or [[non-decreasing function]], then it is even possible to conclude that <math>\sup f(S) = f(\sup S).</math> 
This may be applied, for instance, to conclude that whenever <math>g</math> is a real (or [[Complex number|complex]]) valued function with domain <math>\Omega \neq \varnothing</math> whose [[sup norm]] <math>\|g\|_\infty \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \sup_{x \in \Omega} |g(x)|</math> is finite, then for every non-negative real number <math>q,</math>
<math display=block>\|g\|_\infty^q ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\sup_{x \in \Omega} |g(x)|\right)^q = \sup_{x \in \Omega} \left(|g(x)|^q\right)</math>
since the map <math>f : [0, \infty) \to \R</math> defined by <math>f(x) = x^q</math> is a continuous non-decreasing function whose domain <math>[0, \infty)</math> always contains <math>S := \{|g(x)| : x \in \Omega\}</math> and <math>\sup S \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \|g\|_\infty.</math>

Although this discussion focused on <math>\sup,</math> similar conclusions can be reached for <math>\inf</math> with appropriate changes (such as requiring that <math>f</math> be non-increasing rather than non-decreasing). Other [[Norm (mathematics)|norms]] defined in terms of <math>\sup</math> or <math>\inf</math> include the [[weak Lp space|weak <math>L^{p,w}</math> space]] norms (for <math>1 \leq p < \infty</math>), the norm on [[Lp space|Lebesgue space]] <math>L^\infty(\Omega, \mu),</math> and [[operator norm]]s. Monotone sequences in <math>S</math> that converge to <math>\sup S</math> (or to <math>\inf S</math>) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.