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- Nested_intervals abstract "In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbersInsuch that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of Infor all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In. Without any further information, all that can be said is that the intersection J of all the In, i.e. the set of all points common to the intervals, is either the empty set, a point, or some interval.The possibility of an empty intersection can be illustrated by the intersection when In is the open interval(0, 2−n).Here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2−n.The situation is different for closed intervals. The nested intervals theorem states that if each In is a closed and bounded interval, sayIn = [an, bn]withan ≤ bnthen under the assumption of nesting, the intersection of the In is not empty. It may be a singleton set {c}, or another closed interval [a, b]. More explicitly, the requirement of nesting means that an ≤ an + 1and bn ≥ bn + 1.Moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton.One can consider the complement of each interval, written as . By De Morgan's laws, the complement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there must be something between them. This shows that the intersection of (even an uncountable number of) nested, closed, and bounded intervals is nonempty.".
- Nested_intervals thumbnail Illustration_nested_intervals.svg?width=300.
- Nested_intervals wikiPageID "7532405".
- Nested_intervals wikiPageRevisionID "593009025".
- Nested_intervals hasPhotoCollection Nested_intervals.
- Nested_intervals subject Category:Sets_of_real_numbers.
- Nested_intervals subject Category:Theorems_in_real_analysis.
- Nested_intervals type Abstraction100002137.
- Nested_intervals type Collection107951464.
- Nested_intervals type Communication100033020.
- Nested_intervals type Group100031264.
- Nested_intervals type Message106598915.
- Nested_intervals type Proposition106750804.
- Nested_intervals type Set107996689.
- Nested_intervals type SetsOfRealNumbers.
- Nested_intervals type Statement106722453.
- Nested_intervals type Theorem106752293.
- Nested_intervals type TheoremsInRealAnalysis.
- Nested_intervals comment "In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbersInsuch that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of Infor all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In.".
- Nested_intervals label "Intervallschachtelung".
- Nested_intervals label "Nested intervals".
- Nested_intervals label "Principio de los intervalos encajados".
- Nested_intervals label "Teorema do encaixe de intervalos".
- Nested_intervals label "Лемма о вложенных отрезках".
- Nested_intervals sameAs Intervallschachtelung.
- Nested_intervals sameAs Principio_de_los_intervalos_encajados.
- Nested_intervals sameAs Teorema_do_encaixe_de_intervalos.
- Nested_intervals sameAs m.0264kyh.
- Nested_intervals sameAs Q1638245.
- Nested_intervals sameAs Q1638245.
- Nested_intervals sameAs Nested_intervals.
- Nested_intervals wasDerivedFrom Nested_intervals?oldid=593009025.
- Nested_intervals depiction Illustration_nested_intervals.svg.
- Nested_intervals isPrimaryTopicOf Nested_intervals.