Nested intervals

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In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers

I_n

such that each set I_n is an interval of the real line, for n = 1, 2, 3, ... , and that further

I_{n + 1} is a subset of I_n

for all n. In 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 I_n. Without any further information, all that can be said is that the intersection J of all the I_n, 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 I_n is the open interval

(0, 2⁻ⁿ)

Here the intersection is empty, because no number x is both > 0 and less than every fraction 2⁻ⁿ.

The situation is different for closed intervals. The nested intervals theorem states that if each I_n is a closed interval, say

I_n = [a_n, b_n]

with

a_n < b_n

then under the assumption of nesting, the intersection of the I_n is not empty. It may be a singleton set {c}, or another closed interval [a, b]. More explicitly, the requirement of nesting means that

a_n≤ a_{n + 1}

and

b_n ≥ a_{n + 1}.

The sequences of endpoints respectively form an increasing sequence of real numbers bounded above, and a decreasing sequence bounded below. They therefore tend to respective limits a and b. It can be shown, as this suggests, that either a < b and the intersection is the interval [a, b]]; or that a = b and the intersection consists of the singleton {c}, where c is their common value.

[edit] Higher dimensions

In two dimensions there is a similar result: nested closed disks in the plane must have a common intersection (a disk or single point). This result was shown by Hermann Weyl to classify the singular behaviour of certain differential equations.