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| {{Unreferenced|date=December 2009}}
| | Oscar is how he's known as and he totally loves this name. South Dakota is exactly where me and my spouse live. What I adore doing is to gather badges but I've been taking on new things lately. Hiring is her working day job now and she will not change it anytime quickly.<br><br>Here is my page ... [http://www.youporntime.com/user/KBiaggini std testing at home] |
| [[File:Illustration nested intervals.svg|thumb|400px|right]]
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| In [[mathematics]], a sequence of '''nested intervals''' is understood as a collection of sets of real numbers
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| :''I''<sub>''n''</sub>
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| such that each set ''I''<sub>''n''</sub> is an [[interval of the real line]], for ''n'' = 1, 2, 3, ... , and that further
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| :''I''<sub>''n'' + 1 </sub> is a subset of ''I''<sub>''n''</sub>
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| for 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.
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| The main question to be posed is the nature of the [[intersection (set theory)|intersection]] of all the ''I''<sub>''n''</sub>. Without any further information, all that can be said is that the intersection ''J'' of all the ''I''<sub>''n''</sub>, i.e. the set of all points common to the intervals, is either the [[empty set]], a point, or some interval.
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| The possibility of an empty intersection can be illustrated by the intersection when ''I''<sub>''n''</sub> is the [[open interval]]
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| :(0, 2<sup>−''n''</sup>).
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| Here the intersection is empty, because no number ''x'' is both greater than 0 and less than every fraction 2<sup>−''n''</sup>.
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| The situation is different for [[closed interval]]s. The ''nested intervals theorem'' states that if each ''I''<sub>''n''</sub> is a closed and bounded interval, say
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| :''I''<sub>''n''</sub> = [''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>]
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| with
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| :''a''<sub>''n''</sub> ≤ ''b''<sub>''n''</sub> | |
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| then under the assumption of nesting, the intersection of the ''I''<sub>''n''</sub> is not empty. It may be a singleton set {''c''}, or another closed interval [''a'', ''b'']. More explicitly, the requirement of nesting means that
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| : ''a''<sub>''n''</sub> ≤ ''a''<sub>''n'' + 1</sub>
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| and
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| : ''b''<sub>''n''</sub> ≥ ''b''<sub>''n'' + 1</sub>.
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| Moreover, if the length of the intervals converges to 0, then the intersection of the ''I''<sub>''n''</sub> is a singleton.
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| One can consider the complement of each interval, written as <math>(-\infty,a_n) \cup (b_n, \infty)</math>. 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.
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| ==Higher dimensions==
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| In two dimensions there is a similar result: nested [[closed disk]]s in the plane must have a common intersection. This result was shown by [[Hermann Weyl]] to classify the singular behaviour of certain [[differential equation]]s.
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| ==See also==
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| *[[Bisection]]
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| *[[Cantor's Intersection Theorem]]
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| {{DEFAULTSORT:Nested Intervals}}
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| [[Category:Sets of real numbers]]
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| [[Category:Theorems in real analysis]]
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Oscar is how he's known as and he totally loves this name. South Dakota is exactly where me and my spouse live. What I adore doing is to gather badges but I've been taking on new things lately. Hiring is her working day job now and she will not change it anytime quickly.
Here is my page ... std testing at home