In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 real ...
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 (mathematics), set. A poset consists of a set toget ...
has the least-upper-bound property if every non-empty
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of with an
upper bound has a
''least'' upper bound (supremum) in . Not every (partially) ordered set has the least upper bound property. For example, the set
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 ration ...
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 lin ...
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, such as the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
, the
Bolzano–Weierstrass theorem, 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> suc ...
, and the
Heine–Borel theorem. It is usually taken as an axiom in synthetic
constructions of the real numbers, and it is also intimately related to the construction of the real numbers using
Dedekind cuts.
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 intr ...
, 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 (mathematics), set. A poset consists of a set toget ...
. A
linearly ordered set 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.
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 real ...
s.
* A real number is called an
upper bound for if for all .
* A real number is the least upper bound (or
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
) 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
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 (mathematics), set. A poset consists of a set toget ...
, 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 ration ...
s does not have the least-upper-bound property under the usual order. For instance, the set
:
has an upper bound in , but does not have a least upper bound in (since the square root of two is
irrational). The
construction of the real numbers using
Dedekind cuts 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 lin ...
, such as the convergence of
Cauchy sequences 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 ne ...
. The logical status of the property depends on the
construction of the real numbers 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 eve ...
); 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 th ...
, 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 t ...
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.
Intermediate value theorem
Let be a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
, and suppose that and . In this case, the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
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 sur ...
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 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. This theorem can be proved by considering the set
:
Clearly,
, and is not empty.
In addition, is an upper bound for , so has a least upper bound .
Then must be a
limit point of the sequence , and it follows that has a subsequence that converges to .
Extreme value theorem
Let be a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
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> suc ...
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 suf ...
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 liber ...
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 fu ...
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