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.

Technology Types

collectionmathematical theorempropositionsetsets of real numberstatementtheoremtheorems in analysistheorems in real analysistheory

Synonyms

Nested interval propertyNested Intervals Theorem

Translations

Intervallschachtelung (de)Principio de los intervalos encajados (es)Teorema do encaixe de intervalos (pt)Лема про вкладені відрізки (uk)Лемма о вложенных отрезках (ru)축소구간정리 (ko)