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In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. More generally, a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
has the least-upper-bound property if every non-empty subset of with an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...
has a ''least'' upper bound (supremum) in . Not every (partially) ordered set has the least upper bound property. For example, the set \mathbb of all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s with its natural order does ''not'' have the least upper bound property. The least-upper-bound property is one form of the
completeness axiom Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number li ...
for the real numbers, and is sometimes referred to as Dedekind completeness.Willard says that an ordered space "X is Dedekind complete if every subset of X having an upper bound has a least upper bound." (pp. 124-5, Problem 17E.) It can be used to prove many of the fundamental results of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, such as the intermediate value theorem, the
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
, the
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> su ...
, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic
constructions of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
, and it is also intimately related to the construction of the real numbers using
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
s. In
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
, this property can be generalized to a notion of completeness for any
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
. A
linearly ordered set Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
that is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
and has the least upper bound property is called a
linear continuum In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any t ...
.


Statement of the property


Statement for real numbers

Let be a non-empty set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. * A real number is called an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...
for if for all . * A real number is the least upper bound (or supremum) for if is an upper bound for and for every upper bound of . The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in ''real numbers''.


Generalization to ordered sets

More generally, one may define upper bound and least upper bound for any subset of a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
, with “real number” replaced by “element of ”. In this case, we say that has the least-upper-bound property if every non-empty subset of with an upper bound has a least upper bound in . For example, the set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s does not have the least-upper-bound property under the usual order. For instance, the set : \left\ = \mathbf \cap \left(-\sqrt, \sqrt\right) has an upper bound in , but does not have a least upper bound in (since the square root of two is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
). The
construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
using
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
s takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.


Proof


Logical status

The least-upper-bound property is equivalent to other forms of the
completeness axiom Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number li ...
, such as the convergence of
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s or the
nested intervals theorem In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots as an index. In order for a sequence of intervals to be considered ...
. The logical status of the property depends on the
construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see
least upper bound axiom In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if ev ...
); in a constructive approach, the property must be proved as a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
, either directly from the construction or as a consequence of some other form of completeness.


Proof using Cauchy sequences

It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let be a
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
set of real numbers. If has exactly one element, then its only element is a least upper bound. So consider with more than one element, and suppose that has an upper bound . Since is nonempty and has more than one element, there exists a real number that is not an upper bound for . Define sequences and recursively as follows: # Check whether is an upper bound for . # If it is, let and let . # Otherwise there must be an element in so that . Let and let . Then and as . It follows that both sequences are Cauchy and have the same limit , which must be the least upper bound for .


Applications

The least-upper-bound property of can be used to prove many of the main foundational theorems in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
.


Intermediate value theorem

Let be a continuous function, and suppose that and . In this case, the intermediate value theorem states that must have a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
in the interval . This theorem can be proved by considering the set :. That is, is the initial segment of that takes negative values under . Then is an upper bound for , and the least upper bound must be a root of .


Bolzano–Weierstrass theorem

The
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
for states that every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of real numbers in a closed interval must have a convergent
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
. This theorem can be proved by considering the set : Clearly, a\in S, and is not empty. In addition, is an upper bound for , so has a least upper bound . Then must be a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of the sequence , and it follows that has a subsequence that converges to .


Extreme value theorem

Let be a continuous function and let , where if has no upper bound. The
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> su ...
states that is finite and for some . This can be proved by considering the set :. By definition of , , and by its own definition, is bounded by . If is the least upper bound of , then it follows from continuity that .


Heine–Borel theorem

Let be a closed interval in , and let be a collection of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s that covers . Then the Heine–Borel theorem states that some finite subcollection of covers as well. This statement can be proved by considering the set :. The set obviously contains , and is bounded by by construction. By the least-upper-bound property, has a least upper bound . Hence, is itself an element of some open set , and it follows for that can be covered by finitely many for some sufficiently small . This proves that and is not an upper bound for . Consequently, .


History

The importance of the least-upper-bound property was first recognized by
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his lib ...
in his 1817 paper ''Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewäahren, wenigstens eine reelle Wurzel der Gleichung liege''.


See also

*
List of real analysis topics This is a list of articles that are considered real analysis topics. General topics Limits *Limit of a sequence ** Subsequential limit – the limit of some subsequence *Limit of a function (''see List of limits for a list of limits of common f ...


Notes


References

* * * * * * * *{{cite book , last=Willard , first=Stephen , title=General Topology , isbn=9780486434797 , orig-year=1970, year=2004 , location=Mineola, N.Y. , publisher=Dover Publications Real analysis Order theory Articles containing proofs