Real Numbers:Suprema and Infima - Revision history

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[[Image:Infimum illustration.svg|thumb|right|250px|A set <math>P</math> of real numbers (hollow and filled circles), a subset <math>S</math> of <math>P</math> (filled circles), and the infimum of <math>S.</math> Note that for finite, totally ordered sets the infimum and the [[minimum]] are equal.]]
[[Image:Supremum illustration.svg|thumb|right|250px|A set <math>A</math> of real numbers (blue circles), a set of upper bounds of <math>A</math> (red diamond and circles), and the smallest such upper bound, that is, the supremum of <math>A</math> (red diamond).]]

In [[mathematics]], the '''infimum''' (abbreviated '''inf'''; plural '''infima''') of a [[subset]] <math>S</math> of a [[partially ordered set]] <math>P</math> is a [[greatest element]] in <math>P</math> that is less than or equal to all elements of <math>S,</math> if such an element exists.<ref name=BabyRudin>{{cite book|first=Walter|last=Rudin|author-link=Walter Rudin|title=Principles of Mathematical Analysis|publisher=McGraw-Hill|edition=3rd|year=1976|isbn=0-07-054235-X|chapter="Chapter 1 The Real and Complex Number Systems"|format=print|page=[https://archive.org/details/principlesofmath00rudi/page/n15 4]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref> Consequently, the term ''greatest lower bound'' (abbreviated as {{em|GLB}}) is also commonly used.<ref name=BabyRudin />

The '''supremum''' (abbreviated '''sup'''; plural '''suprema''') of a subset <math>S</math> of a partially ordered set <math>P</math> is the [[least element]] in <math>P</math> that is greater than or equal to all elements of <math>S,</math> if such an element exists.<ref name=BabyRudin /> Consequently, the supremum is also referred to as the ''least upper bound'' (or {{em|LUB}}).<ref name=BabyRudin />

The infimum is in a precise sense [[Duality (order theory)|dual]] to the concept of a supremum. Infima and suprema of [[real number]]s are common special cases that are important in [[Mathematical analysis|analysis]], and especially in [[Lebesgue integration]]. However, the general definitions remain valid in the more abstract setting of [[order theory]] where arbitrary partially ordered sets are considered.

The concepts of infimum and supremum are similar to [[minimum]] and [[maximum]], but are more useful in analysis because they better characterize special sets which may have {{em|no minimum or maximum}}. For instance, the set of [[positive real numbers]] <math>\R^+</math> (not including <math>0</math>) does not have a minimum, because any given element of <math>\R^+</math> could simply be divided in half resulting in a smaller number that is still in <math>\R^+.</math> There is, however, exactly one infimum of the positive real numbers: <math>0,</math> which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.

== Formal definition ==

[[File:Illustration of supremum.svg|thumb|supremum = least upper bound]]
A {{em|lower bound}} of a subset <math>S</math> of a partially ordered set <math>(P, \leq)</math> is an element <math>a</math> of <math>P</math> such that
* <math>a \leq x</math> for all <math>x \in S.</math>
A lower bound <math>a</math> of <math>S</math> is called an {{em|infimum}} (or {{em|greatest lower bound}}, or {{em|meet}}) of <math>S</math> if
* for all lower bounds <math>y</math> of <math>S</math> in <math>P,</math> <math>y \leq a</math> (<math>a</math> is larger than or equal to any other lower bound).

Similarly, an {{em|upper bound}} of a subset <math>S</math> of a partially ordered set <math>(P, \leq)</math> is an element <math>b</math> of <math>P</math> such that
* <math>b \geq x</math> for all <math>x \in S.</math>
An upper bound <math>b</math> of <math>S</math> is called a {{em|supremum}} (or {{em|least upper bound}}, or {{em|join}}) of <math>S</math> if
* for all upper bounds <math>z</math> of <math>S</math> in <math>P,</math> <math>z \geq b</math> (<math>b</math> is less than or equal to any other upper bound).

== Existence and uniqueness ==

Infima and suprema do not necessarily exist. Existence of an infimum of a subset <math>S</math> of <math>P</math> can fail if <math>S</math> has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique.

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a [[Lattice (order)|lattice]] is a partially ordered set in which all {{em|nonempty finite}} subsets have both a supremum and an infimum, and a [[complete lattice]] is a partially ordered set in which {{em|all}} subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on [[Completeness (order theory)|completeness properties]].

If the supremum of a subset <math>S</math> exists, it is unique. If <math>S</math> contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to <math>S</math> (or does not exist). Likewise, if the infimum exists, it is unique. If <math>S</math> contains a least element, then that element is the infimum; otherwise, the infimum does not belong to <math>S</math> (or does not exist).

== Relation to maximum and minimum elements ==

The infimum of a subset <math>S</math> of a partially ordered set <math>P,</math> assuming it exists, does not necessarily belong to <math>S.</math> If it does, it is a [[Minimal element|minimum or least element]] of <math>S.</math> Similarly, if the supremum of <math>S</math> belongs to <math>S,</math> it is a [[Maximal element|maximum or greatest element]] of <math>S.</math>

For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number <math>x,</math> there is another negative real number <math>\tfrac{x}{2},</math> which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, <math>0</math> is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

However, the definition of [[Maximal element|maximal and minimal elements]] is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.

=== Minimal upper bounds ===
Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a [[Totally ordered set|total]] one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, let <math>S</math> be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from <math>S</math> together with the set of [[integer]]s <math>\Z</math> and the set of positive real numbers <math>\R^+,</math> ordered by subset inclusion as above. Then clearly both <math>\Z</math> and <math>\R^+</math> are greater than all finite sets of natural numbers. Yet, neither is <math>\R^+</math> smaller than <math>\Z</math> nor is the converse true: both sets are minimal upper bounds but none is a supremum.

=== 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>p,</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{S}</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.

== Infima and suprema of real numbers ==

In [[Mathematical analysis|analysis]], infima and suprema of subsets <math>S</math> of the [[real numbers]] are particularly important. For instance, the negative [[real number]]s do not have a greatest element, and their supremum is <math>0</math> (which is not a negative real number).<ref name=BabyRudin />
The [[completeness of the real numbers]] implies (and is equivalent to) that any bounded nonempty subset <math>S</math> of the real numbers has an infimum and a supremum. If <math>S</math> is not bounded below, one often formally writes <math>\inf_{} S = -\infty.</math> If <math>S</math> is [[Empty set|empty]], one writes <math>\inf_{} S = +\infty.</math>

=== Properties ===

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets <math>A, B \subseteq \R,</math> and scalar <math>r \in \R.</math> Define

* <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}}
* <math>r A = \{ r \cdot a : a \in A \}</math>; the scalar product of a set is just the scalar multiplied by every element in the set.
* <math>A + B = \{ a + b : a \in A, b \in B \}</math>; called the [[Minkowski sum]], it is the arithmetic sum of two sets is the sum of all possible pairs of numbers, one from each set.
* <math>A \cdot B = \{ a \cdot b : a \in A, b \in B \}</math>; the arithmetic product of two sets is all products of pairs of elements, one from each set.
* If <math>\varnothing \neq S \subseteq \R</math> then there exists a sequence <math>s_{\bull} = \left(s_n\right)_{n=1}^{\infty}</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \sup S.</math> Similarly, there will exist a (possibly different) sequence <math>s_{\bull}</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \inf S.</math> Consequently, if the limit <math>\lim_{n \to \infty} s_n = \sup S</math> is a real number and if <math>f : \R \to X</math> is a continuous function, then <math>f\left(\sup S\right)</math> is necessarily an [[adherent point]] of <math>f(S).</math>

In those cases where the infima and suprema of the sets <math>A</math> and <math>B</math> exist, the following identities hold:
* <math>p = \inf A</math> if and only <math>p</math> is a [[Minorant]] and for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon < p + \epsilon.</math>
* <math>p = \sup A</math> if and only <math>p</math> is a [[Majorant]] and if for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon > p - \epsilon</math>
* If <math>A \subseteq B</math> and then <math>\inf A \geq \inf B</math> and <math>\sup A \leq \sup B.</math>

* If <math>r > 0</math> then <math>\inf (r \cdot A) = r \left(\inf A\right)</math> and <math>\sup (r \cdot A) = r \left(\sup A\right).</math>
* If <math>r \leq 0</math> then <math>\inf (r \cdot A) = r \left(\sup A\right)</math> and <math>\sup (r \cdot A) = r \left(\inf A\right).</math>
* <math>\inf (A + B) = \left(\inf A\right) + \left(\inf B\right)</math> and <math>\sup (A + B) = \left(\sup A\right) + \left(\sup B\right).</math>
* If <math>A</math> and <math>B</math> are nonempty sets of positive real numbers then <math>\inf (A \cdot B) = \left(\inf A\right) \cdot \left(\inf B\right)</math> and similarly for suprema <math>\sup (A \cdot B) = \left(\sup A\right) \cdot \left(\sup B\right).</math><ref name="zakon">{{cite book|title=Mathematical Analysis I|first=Elias|last=Zakon|pages=39–42|publisher=Trillia Group|date=2004|url=http://www.trillia.com/zakon-analysisI.html}}</ref>
* If <math>S \subseteq (0, \infty)</math> is non-empty and if <math>\frac{1}{S} := \left\{ \frac{1}{s} : s \in S \right\},</math> then <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used.<ref group="note" name="DivisionByInfinityOr0">The definition <math>\frac{1}{\infty} := 0</math> is commonly used with the [[extended real number]]s; in fact, with this definition the equality <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\frac{1}{0}</math> is usually left undefined, which is why the equality <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math></ref> This equality may alternatively be written as <math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \frac{1}{s}.</math> Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \frac{1}{S} = \infty,</math> where if<ref group=note name="DivisionByInfinityOr0" /> <math>\inf_{} S > 0,</math> then <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}.</math>

== Examples ==

=== Infima ===

* The infimum of the set of numbers <math>\{2, 3, 4\}</math> is <math>2.</math> The number <math>1</math> is a lower bound, but not the greatest lower bound, and hence not the infimum.
* More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the [[minimum]] of the set.
* <math>\inf \{ 1, 2, 3, \ldots \} = 1.</math>
* <math>\inf \{ x \in \R : 0 < x < 1 \} = 0.</math>
* <math>\inf \left\{ x \in \Q : x^3 > 2 \right\} = \sqrt[3]{2}.</math>
* <math>\inf \left\{ (-1)^n + \tfrac{1}{n} : n = 1, 2, 3, \ldots \right\} = -1.</math>
* If <math>\left(x_n\right)_{n=1}^{\infty}</math> is a decreasing sequence with limit <math>x,</math> then <math>\inf x_n = x.</math>

=== 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 subset <math>S</math> of <math>(P, \subseteq),</math> where <math>P</math> is the [[power set]] of some set, is the supremum with respect to <math>\,\subseteq\,</math> (subset) of a subset <math>S</math> of <math>P</math> is the [[Union (set theory)|union]] of the elements of <math>S.</math>

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 The supremum of a subset <math>S</math> of <math>(P, \subseteq),</math> where <math>P</math> is the power set of some set, is the supremum with respect to <math>\,\subseteq\,</math> (subset) of a subset <math>S</math> of <math>P</math> is the union of the elements of <math>S.</math>
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Infimum_and_supremum Infimum and supremum], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 21:09, 19 October 2021
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	The supremum of a subset <math>S</math> of <math>(P, \subseteq),</math> where <math>P</math> is the power set of some set, is the supremum with respect to <math>\,\subseteq\,</math> (subset) of a subset <math>S</math> of <math>P</math> is the union of the elements of <math>S.</math>		The supremum of a subset <math>S</math> of <math>(P, \subseteq),</math> where <math>P</math> is the power set of some set, is the supremum with respect to <math>\,\subseteq\,</math> (subset) of a subset <math>S</math> of <math>P</math> is the union of the elements of <math>S.</math>
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Infimum_and_supremum Infimum and supremum], Wikipedia

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 * <math>\inf (A + B) = \left(\inf A\right) + \left(\inf B\right)</math> and <math>\sup (A + B) = \left(\sup A\right) + \left(\sup B\right).</math>
 * If <math>A</math> and <math>B</math> are nonempty sets of positive real numbers then <math>\inf (A \cdot B) = \left(\inf A\right) \cdot \left(\inf B\right)</math> and similarly for suprema <math>\sup (A \cdot B) = \left(\sup A\right) \cdot \left(\sup B\right).</math>
-* If <math>S \subseteq (0, \infty)</math> is non-empty and if <math>\frac{1}{S} := \left\{ \frac{1}{s} : s \in S \right\},</math> then <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used.<ref group="note" name="DivisionByInfinityOr0">The definition <math>\frac{1}{\infty} := 0</math> is commonly used with the extended real numbers; in fact, with this definition the equality <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\frac{1}{0}</math> is usually left undefined, which is why the equality <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math></ref> This equality may alternatively be written as <math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \frac{1}{s}.</math> Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \frac{1}{S} = \infty,</math> where if <math>\inf_{} S > 0,</math> then <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}.</math>
+* If <math>S \subseteq (0, \infty)</math> is non-empty and if <math>\frac{1}{S} := \left\{ \frac{1}{s} : s \in S \right\},</math> then <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used. The definition <math>\frac{1}{\infty} := 0</math> is commonly used with the extended real numbers; in fact, with this definition the equality <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\frac{1}{0}</math> is usually left undefined, which is why the equality <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math> This equality may alternatively be written as <math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \frac{1}{s}.</math> Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \frac{1}{S} = \infty,</math> where if <math>\inf_{} S > 0,</math> then <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}.</math>
 == Examples ==

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 [[Image:Supremum illustration.svg|thumb|right|250px|A set <math>A</math> of real numbers (blue circles), a set of upper bounds of <math>A</math> (red diamond and circles), and the smallest such upper bound, that is, the supremum of <math>A</math> (red diamond).]]
-In [[mathematics]], the '''infimum''' (abbreviated '''inf'''; plural '''infima''') of a [[subset]] <math>S</math> of a [[partially ordered set]] <math>P</math> is a [[greatest element]] in <math>P</math> that is less than or equal to all elements of <math>S,</math> if such an element exists.<ref name=BabyRudin>{{cite book|first=Walter|last=Rudin|author-link=Walter Rudin|title=Principles of Mathematical Analysis|publisher=McGraw-Hill|edition=3rd|year=1976|isbn=0-07-054235-X|chapter="Chapter 1 The Real and Complex Number Systems"|format=print|page=[https://archive.org/details/principlesofmath00rudi/page/n15 4]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref>  Consequently, the term ''greatest lower bound'' (abbreviated as {{em|GLB}}) is also commonly used.<ref name=BabyRudin />
+In mathematics, the '''infimum''' (abbreviated '''inf'''; plural '''infima''') of a subset <math>S</math> of a partially ordered set <math>P</math> is a greatest element in <math>P</math> that is less than or equal to all elements of <math>S,</math> if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as {{em|GLB}}) is also commonly used.
-The '''supremum''' (abbreviated '''sup'''; plural '''suprema''') of a subset <math>S</math> of a partially ordered set <math>P</math> is the [[least element]] in <math>P</math> that is greater than or equal to all elements of <math>S,</math> if such an element exists.<ref name=BabyRudin /> Consequently, the supremum is also referred to as the ''least upper bound'' (or {{em|LUB}}).<ref name=BabyRudin />
+The '''supremum''' (abbreviated '''sup'''; plural '''suprema''') of a subset <math>S</math> of a partially ordered set <math>P</math> is the [[least element]] in <math>P</math> that is greater than or equal to all elements of <math>S,</math> if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or {{em|LUB}}).
 The infimum is in a precise sense [[Duality (order theory)|dual]] to the concept of a supremum.  Infima and suprema of [[real number]]s are common special cases that are important in [[Mathematical analysis|analysis]], and especially in [[Lebesgue integration]].  However, the general definitions remain valid in the more abstract setting of [[order theory]] where arbitrary partially ordered sets are considered.

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 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>
+<math>\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>