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In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such 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 In 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. 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. with and

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dbo:abstract
  • Лема про вкладені відрізки (uk)
  • Das Intervallschachtelungsprinzip wird besonders in der Analysis in Beweisen benutzt und bildet in der numerischen Mathematik die Grundlage für einige Lösungsverfahren. Das Prinzip ist Folgendes: Man fängt mit einem beschränkten Intervall an und wählt aus diesem Intervall ein abgeschlossenes Intervall, das komplett in dem vorherigen Intervall liegt, wählt dort wieder ein abgeschlossenes Intervall heraus und so weiter. Werden die Längen der Intervalle beliebig klein, konvergiert also ihre Länge gegen Null, so gibt es genau eine reelle Zahl, die in allen Intervallen enthalten ist. Wegen dieser Eigenschaft können Intervallschachtelungen herangezogen werden, um mit ihnen die reellen Zahlen als Zahlbereichserweiterung der rationalen Zahlen zu konstruieren. Grundideen in Form des Arguments der vollständigen Teilung finden sich bereits bei Zenon von Elea und Aristoteles. (de)
  • 수학에서, 축소구간열(縮小區間列, sequence of nested intervals)은 각 구간이 바로 앞 구간의 부분 집합인 구간들의 열이다. 축소구간정리(縮小區間定理, 영어: nested intervals theorem)에 따르면, 닫힌구간으로 구성된 축소구간열은 적어도 하나의 공통 원소를 갖는다. (ko)
  • In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such 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 In 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. 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, say In = [an, bn] with an ≤ bn then 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 + 1 and 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. (en)
  • Em matemática, o teorema do encaixe de intervalos afirma que qualquer sucessão decrescente de intervalos de números reais tem, pelo menos, um ponto em comum. (pt)
  • 在數學中,一串區間套是實數中的一串區間In(n=1, 2, 3, ...),使得對於每個n都有In + 1 是In的子集,有時我們要求它是真子集。換而言之,在這串區間中,區間從左邊逐漸往右收縮,而在右邊逐漸往左收縮。 關於區間套的主要問題在於探討所有區間In的交集(記作J)的性狀。 事實上,當In都是開集時,J有可能為空集。例如開區間套(0, 2−n的交集就是空集:任何一個正數x都在n充分大之後大於2−n,故而x不在J中。 但對於閉集而言,情況有所不同。事實上,我們有閉區間套定理,這一定理刻劃了實數的完備性。定理聲稱對於任一的有界閉區間套In(例如In = [an, bn]並滿足an ≤ bn),它們的交集In非空,且為閉區間;特別地,假若,則它們的交集J為一個包含且僅包含的單點集。 (zh)
  • Лемма о вложенных отрезках, или принцип вложенных отрезков Коши — Кантора, или принцип непрерывности Кантора — фундаментальное утверждение в математическом анализе, связанное с полнотой поля вещественных чисел. (ru)
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  • Лема про вкладені відрізки (uk)
  • 수학에서, 축소구간열(縮小區間列, sequence of nested intervals)은 각 구간이 바로 앞 구간의 부분 집합인 구간들의 열이다. 축소구간정리(縮小區間定理, 영어: nested intervals theorem)에 따르면, 닫힌구간으로 구성된 축소구간열은 적어도 하나의 공통 원소를 갖는다. (ko)
  • Em matemática, o teorema do encaixe de intervalos afirma que qualquer sucessão decrescente de intervalos de números reais tem, pelo menos, um ponto em comum. (pt)
  • 在數學中,一串區間套是實數中的一串區間In(n=1, 2, 3, ...),使得對於每個n都有In + 1 是In的子集,有時我們要求它是真子集。換而言之,在這串區間中,區間從左邊逐漸往右收縮,而在右邊逐漸往左收縮。 關於區間套的主要問題在於探討所有區間In的交集(記作J)的性狀。 事實上,當In都是開集時,J有可能為空集。例如開區間套(0, 2−n的交集就是空集:任何一個正數x都在n充分大之後大於2−n,故而x不在J中。 但對於閉集而言,情況有所不同。事實上,我們有閉區間套定理,這一定理刻劃了實數的完備性。定理聲稱對於任一的有界閉區間套In(例如In = [an, bn]並滿足an ≤ bn),它們的交集In非空,且為閉區間;特別地,假若,則它們的交集J為一個包含且僅包含的單點集。 (zh)
  • Лемма о вложенных отрезках, или принцип вложенных отрезков Коши — Кантора, или принцип непрерывности Кантора — фундаментальное утверждение в математическом анализе, связанное с полнотой поля вещественных чисел. (ru)
  • Das Intervallschachtelungsprinzip wird besonders in der Analysis in Beweisen benutzt und bildet in der numerischen Mathematik die Grundlage für einige Lösungsverfahren. Das Prinzip ist Folgendes: Man fängt mit einem beschränkten Intervall an und wählt aus diesem Intervall ein abgeschlossenes Intervall, das komplett in dem vorherigen Intervall liegt, wählt dort wieder ein abgeschlossenes Intervall heraus und so weiter. Werden die Längen der Intervalle beliebig klein, konvergiert also ihre Länge gegen Null, so gibt es genau eine reelle Zahl, die in allen Intervallen enthalten ist. Wegen dieser Eigenschaft können Intervallschachtelungen herangezogen werden, um mit ihnen die reellen Zahlen als Zahlbereichserweiterung der rationalen Zahlen zu konstruieren. (de)
  • In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such 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 In 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. 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. with and (en)
rdfs:label
  • Intervallschachtelung (de)
  • Principio de los intervalos encajados (es)
  • 축소구간정리 (ko)
  • Nested intervals (en)
  • Лемма о вложенных отрезках (ru)
  • Teorema do encaixe de intervalos (pt)
  • 区间套 (zh)
  • Лема про вкладені відрізки (uk)
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