Relatively hyperbolic group

2011-04-17T23:32:00Z

24.136.14.21: /* Formal definition */

In [[mathematics]], the '''least-upper-bound property''' (sometimes '''supremum property of the real numbers''') is a fundamental property of the [[real number]]s and certain other ordered sets. The property states that any non-empty [[set (mathematics)|set]] of real numbers that has an [[upper bound]] necessarily has a [[least upper bound]] (or supremum).

The least-upper-bound property is one form of the [[completeness axiom]] for the real numbers, and is sometimes referred to as '''Dedekind completeness'''. It can be used to prove many of the fundamental results of [[real analysis]], such as the [[intermediate value theorem]], the [[Bolzano–Weierstrass theorem]], the [[extreme value theorem]], and the [[Heine–Borel theorem]]. It is usually taken as an axiom in synthetic [[construction of the real numbers|constructions of the real numbers]] (see [[least upper bound axiom]]), and it is also intimately related to the construction of the real numbers using [[Dedekind cut]]s.

In [[order theory]], this property can be generalized to a notion of [[completeness (order theory)|completeness]] for any [[partially ordered set]]. A [[linearly ordered set]] that is [[dense order|dense]] and has the least upper bound property is called a [[linear continuum]].

==Statement of the property==

===Statement for real numbers===
Let {{math|''S''}} be a non-empty set of [[real number]]s.
* A real number {{math|''x''}} is called an '''[[upper bound]]''' for {{math|''S''}} if {{math|''x'' ≥ ''s''}} for all {{math|''s'' ∈ ''S''}}.
* A real number {{math|''x''}} is the '''least upper bound''' (or '''[[supremum]]''') for {{math|''S''}} if {{math|''x''}} is an upper bound for {{math|''S''}} and {{math|''x'' ≤ ''y''}} for every upper bound {{math|''y''}} of {{math|''S''}}.
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===
{{main|Completeness (order theory)}}
More generally, one may define upper bound and least upper bound for any [[subset]] of a [[partially ordered set]] {{math|''X''}}, with “real number” replaced by “element of {{math|''X''}}”. In this case, we say that {{math|''X''}} has the least-upper-bound property if every non-empty subset of {{math|''X''}} with an upper bound has a least upper bound.

For example, the set {{math|'''Q'''}} of [[rational number]]s does not have the least-upper-bound property under the usual order. For instance, the set

: <math> \left(-\sqrt{2}, \sqrt{2}\right) \cap \mathbf{Q} = \left\{ x \in \mathbf{Q} : x^2 \le 2 \right\} \, </math>

has an upper bound in {{math|'''Q'''}}, but does not have a least upper bound in {{math|'''Q'''}} (since the square root of two is [[Irrational number|irrational]]). The [[construction of the real numbers]] using [[Dedekind cut]]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]], such as the convergence of [[Cauchy sequence]]s or the [[nested intervals theorem]]. The logical status of the property depends on the [[construction of the real numbers]] used: in the [[Construction_of_the_real_numbers#Synthetic_approach|synthetic approach]], the property is usually taken as an axiom for the real numbers (see [[least upper bound axiom]]); in a constructive approach, the property must be proved as a [[theorem]], 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 {{math|''S''}} be a [[nonempty]] set of real numbers, and suppose that {{math|''S''}} has an upper bound {{math|''B''1}}. Since {{math|''S''}} is nonempty, there exists a real number {{math|''A''1}} that is not an upper bound for {{math|''S''}}. Define sequences {{math|''A''1, ''A''2, ''A''3, ...}} and {{math|''B''1, ''B''2, ''B''3, ...}} recursively as follows:
# Check whether {{math|(''An'' + ''Bn'') ⁄ 2}} is an upper bound for {{math|''S''}}.
# If it is, let {{math|''A''''n''+1 {{=}} ''An''}} and let {{math|''B''''n''+1 {{=}} (''An'' + ''Bn'') ⁄ 2}}.
# Otherwise there must be an element {{math|''s''}} in {{math|''S''}} so that {{math|''s''>(''An'' + ''Bn'') ⁄ 2}}. Let {{math|''A''''n''+1 {{=}} ''s''}} and let {{math|''B''''n''+1 {{=}} ''Bn''}}.
Then {{math|''A''1 ≤ ''A''2 ≤ ''A''3 ≤ ⋯ ≤ ''B''3 ≤ ''B''2 ≤ ''B''1}} and {{math|{{!}}''An'' − ''Bn''{{!}} → 0}} as {{math|''n'' → ∞}}. It follows that both sequences are Cauchy and have the same limit {{math|''L''}}, which must be the least upper bound for {{math|''S''}}.

==Applications==
The least-upper-bound property of {{math|'''R'''}} can be used to prove many of the main foundational theorems in [[real analysis]].

===Intermediate value theorem===
Let {{math|''f'' : [''a'', ''b''] → '''R'''}} be a [[continuous function]], and suppose that {{math|''f'' (''a'') < 0}} and {{math|''f'' (''b'') > 0}}. In this case, the [[intermediate value theorem]] states that {{math|''f''}} must have a [[Root of a function|root]] in the interval {{math|[''a'', ''b'']}}. This theorem can proved by considering the set
:{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : ''f'' (''x'') < 0 for all ''x'' ≤ ''s''} }}.
That is, {{math|''S''}} is the initial segment of {{math|[''a'', ''b'']}} that takes negative values under {{math|''f''}}. Then {{math|''b''}} is an upper bound for {{math|''S''}}, and the least upper bound must be a root of {{math|''f''}}.

===Bolzano–Weierstrass theorem===
The [[Bolzano–Weierstrass theorem]] for {{math|'''R'''}} states that every [[sequence]] {{math|''xn''}} of real numbers in a closed interval {{math|[''a'', ''b'']}} must have a convergent [[subsequence]]. This theorem can be proved by considering the set
:{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : ''s'' ≤ ''xn'' for infinitely many ''n''} }}.
Clearly {{math|''b''}} is an upper bound for {{math|''S''}}, so {{math|''S''}} has a least upper bound {{math|''c''}}. Then {{math|''c''}} must be a [[limit point]] of the sequence {{math|''xn''}}, and it follows that {{math|''xn''}} has a subsequence that converges to {{math|''c''}}.

===Extreme value theorem===
Let {{math|''f'' : [''a'', ''b''] → '''R'''}} be a [[continuous function]] and let {{math|''M'' {{=}} sup ''f'' ([''a'', ''b''])}}, where {{math|''M'' {{=}} ∞}} if {{math|''f'' ([''a'', ''b''])}} has no upper bound. The [[extreme value theorem]] states that {{math|''M''}} is finite and {{math|''f'' (''c'') {{=}} ''M''}} for some {{math|''c'' ∈ [''a'', ''b'']}}. This can be proved by considering the set
:{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : sup ''f'' ([''s'', ''b'']) {{=}} ''M''} }}.
If {{math|''c''}} is the least upper bound of this set, then it follows from continuity that {{math|''f'' (''c'') {{=}} ''M''}}.

===Heine–Borel theorem===
Let {{math|[''a'', ''b'']}} be a closed interval in {{math|'''R'''}}, and let {{math|{''Uα''} }} be a collection of [[open set]]s that [[Cover (topology)|covers]] {{math|[''a'', ''b'']}}. Then the [[Heine–Borel theorem]] states that some finite subcollection of {{math|{''Uα''} }} covers {{math|[''a'', ''b'']}} as well. This statement can be proved by considering the set
:{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : [''a'', ''s''] can be covered by finitely many ''Uα''} }}.
This set must have a least upper bound {{math|''c''}}. But {{math|''c''}} is itself an element of some open set {{math|''Uα''}}, and it follows that {{math|[''a'', ''c'' + ''δ'']}} can be covered by finitely many {{math|''Uα''}} for some sufficiently small {{math|''δ'' > 0}}. This proves that {{math|''c'' + ''δ'' ∈ ''S''}}, and it also yields a contradiction unless {{math|''c'' {{=}} ''b''}}.

==See also==
* [[List of real analysis topics]]

==References==
*{{cite book
| last = Aliprantis
| first = Charalambos D
| authorlink = Charalambos D. Aliprantis
| coauthors = Burkinshaw, Owen
| title = Principles of real analysis
| edition = Third
| publisher = Academic
| date = 1998
| pages =
| isbn = 0-12-050257-7

}}
*{{cite book |author=Browder, Andrew |title=Mathematical Analysis: An Introduction |series=Undergraduate Texts in Mathematics |location=New York |publisher=Springer-Verlag |date=1996 |isbn=0-387-94614-4 }}

*{{cite book |author=Bartle, Robert G. and Sherbert, Donald R. |title=Introduction to Real Analysis |edition=4 |location=New York |publisher=John Wiley and Sons |date=2011 |isbn=978-0-471-43331-6 |ref=Bartle}}

*{{cite book |author=Abbott, Stephen |title=Understanding Analysis |series=Undergradutate Texts in Mathematics |isbn=0-387-95060-5 |date=2001 |location=New York |publisher=Springer-Verlag }}

*{{cite book |author=Rudin, Walter |title=Principles of Mathematical Analysis |series=Walter Rudin Student Series in Advanced Mathematics |edition=3 |publisher=McGraw–Hill |isbn=978-0-07-054235-8 }}

*{{cite book |author=Dangello, Frank and Seyfried, Michael |title=Introductory Real Analysis |isbn=978-0-395-95933-6 |publisher=Brooks Cole |date=1999 }}

*{{cite book |author=Bressoud, David |title=A Radical Approach to Real Analysis |isbn=0-88385-747-2 |publisher=MAA |date=2007 }}

[[Category:Real analysis]]
[[Category:Order theory]]
[[Category:Articles containing proofs]]

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Relatively hyperbolic group