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| 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).
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| 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.
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| 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]].
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| ==Statement of the property==
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| ===Statement for real numbers===
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| Let {{math|''S''}} be a non-empty set of [[real number]]s.
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| * A real number {{math|''x''}} is called an '''[[upper bound]]''' for {{math|''S''}} if {{math|''x'' ≥ ''s''}} for all {{math|''s'' ∈ ''S''}}.
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| * 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''}}.
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| 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''.
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| ===Generalization to ordered sets===
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| {{main|Completeness (order theory)}}
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| 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.
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| 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
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| : <math> \left(-\sqrt{2}, \sqrt{2}\right) \cap \mathbf{Q} = \left\{ x \in \mathbf{Q} : x^2 \le 2 \right\} \, </math>
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| 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.
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| ==Proof==
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| ===Logical status===
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| 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.
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| ===Proof using Cauchy sequences===
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| 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''<sub>1</sub>}}. Since {{math|''S''}} is nonempty, there exists a real number {{math|''A''<sub>1</sub>}} that is not an upper bound for {{math|''S''}}. Define sequences {{math|''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, ...}} and {{math|''B''<sub>1</sub>, ''B''<sub>2</sub>, ''B''<sub>3</sub>, ...}} recursively as follows:
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| # Check whether {{math|(''A<sub>n</sub>'' + ''B<sub>n</sub>'') ⁄ 2}} is an upper bound for {{math|''S''}}.
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| # If it is, let {{math|''A''<sub>''n''+1</sub> {{=}} ''A<sub>n</sub>''}} and let {{math|''B''<sub>''n''+1</sub> {{=}} (''A<sub>n</sub>'' + ''B<sub>n</sub>'') ⁄ 2}}.
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| # Otherwise there must be an element {{math|''s''}} in {{math|''S''}} so that {{math|''s''>(''A<sub>n</sub>'' + ''B<sub>n</sub>'') ⁄ 2}}. Let {{math|''A''<sub>''n''+1</sub> {{=}} ''s''}} and let {{math|''B''<sub>''n''+1</sub> {{=}} ''B<sub>n</sub>''}}.
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| Then {{math|''A''<sub>1</sub> ≤ ''A''<sub>2</sub> ≤ ''A''<sub>3</sub> ≤ ⋯ ≤ ''B''<sub>3</sub> ≤ ''B''<sub>2</sub> ≤ ''B''<sub>1</sub>}} and {{math|{{!}}''A<sub>n</sub>'' − ''B<sub>n</sub>''{{!}} → 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''}}.
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| ==Applications==
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| The least-upper-bound property of {{math|'''R'''}} can be used to prove many of the main foundational theorems in [[real analysis]].
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| ===Intermediate value theorem===
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| 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
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| :{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : ''f'' (''x'') < 0 for all ''x'' ≤ ''s''} }}.
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| 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''}}.
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| ===Bolzano–Weierstrass theorem===
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| The [[Bolzano–Weierstrass theorem]] for {{math|'''R'''}} states that every [[sequence]] {{math|''x<sub>n</sub>''}} of real numbers in a closed interval {{math|[''a'', ''b'']}} must have a convergent [[subsequence]]. This theorem can be proved by considering the set
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| :{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : ''s'' ≤ ''x<sub>n</sub>'' for infinitely many ''n''} }}.
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| 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|''x<sub>n</sub>''}}, and it follows that {{math|''x<sub>n</sub>''}} has a subsequence that converges to {{math|''c''}}.
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| ===Extreme value theorem===
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| 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
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| :{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : sup ''f'' ([''s'', ''b'']) {{=}} ''M''} }}.
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| If {{math|''c''}} is the least upper bound of this set, then it follows from continuity that {{math|''f'' (''c'') {{=}} ''M''}}.
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| ===Heine–Borel theorem===
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| Let {{math|[''a'', ''b'']}} be a closed interval in {{math|'''R'''}}, and let {{math|{''U<sub>α</sub>''} }} 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|{''U<sub>α</sub>''} }} covers {{math|[''a'', ''b'']}} as well. This statement can be proved by considering the set
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| :{{math|''S'' {{=}} {''s'' ∈ [''a'', ''b''] : [''a'', ''s''] can be covered by finitely many ''U<sub>α</sub>''} }}.
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| This set must have a least upper bound {{math|''c''}}. But {{math|''c''}} is itself an element of some open set {{math|''U<sub>α</sub>''}}, and it follows that {{math|[''a'', ''c'' + ''δ'']}} can be covered by finitely many {{math|''U<sub>α</sub>''}} for some sufficiently small {{math|''δ'' > 0}}. This proves that {{math|''c'' + ''δ'' ∈ ''S''}}, and it also yields a contradiction unless {{math|''c'' {{=}} ''b''}}.
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| ==See also==
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| * [[List of real analysis topics]]
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| ==References==
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| *{{cite book
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| | last = Aliprantis
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| | first = Charalambos D
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| | authorlink = Charalambos D. Aliprantis
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| | coauthors = Burkinshaw, Owen
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| | title = Principles of real analysis
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| | edition = Third
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| | publisher = Academic
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| | date = 1998
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| | pages =
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| | isbn = 0-12-050257-7
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| }}
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| *{{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 }}
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| *{{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}}
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| *{{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 }}
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| *{{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 }}
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| *{{cite book |author=Dangello, Frank and Seyfried, Michael |title=Introductory Real Analysis |isbn=978-0-395-95933-6 |publisher=Brooks Cole |date=1999 }}
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| *{{cite book |author=Bressoud, David |title=A Radical Approach to Real Analysis |isbn=0-88385-747-2 |publisher=MAA |date=2007 }}
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| [[Category:Real analysis]]
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| [[Category:Order theory]]
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| [[Category:Articles containing proofs]]
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It involves expertise and knowledge of various tools and technologies used for creating websites. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts.
Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. If you enjoyed this post and you would like to obtain more details regarding wordpress backup plugin kindly see our own website. Wordpress have every reason with it which promote wordpress development. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. This platform can be customizedaccording to the requirements of the business. The first thing you need to do is to choose the right web hosting plan. After that the developer adds the unordered list for navigations.
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