In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:

1. ).

2. ).

).

Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and circumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi).

The central question to be posed is the nature of the intersection). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers).

Historic motivation

As stated in the introduction, historic users of mathematics discovered the nesting of intervals and closely related algorithms as methods for specific calculations. Some variations and modern interpretations of these ancient techniques will be introduced here:

Computation of square roots

has to be found. If one knows the next higher perfect square.

can now be defined recursively), one can define

with arbitrarily good precision (given enough computational time).

and calculating the reciprocal after the desired level of precision has been acquired.

Example

must certainly found within this interval. Thus, using this interval, one can continue to the next step of the algorithm by calculating the midpoint of the interval, determining whether the square of the midpoint is greater than or less than 19, and setting the boundaries of the next interval accordingly before repeating the process:

1. to be estimated with a greater precision, either by increasing the lower bounds of the interval or decreasing the upper bounds of the interval.

2. This procedure can be repeated as many times as needed to attain the desired level of precision. Theoretically, by repeating the steps indefinitely, one can arrive at the true value of this square root.

Herons method

The Babylonian method given by

1. .

has to be considered, which converges (as does of course the lower interval bound). This algorithm is a special case of Newton's method.

Archimedes' circle measurement

.

Around 250 BCE Archimedes of Syracuse started with regular hexagons.

Around the year 1600 CE, Archimedes' method was still the gold standard for calculating Pi and was used by Dutch mathematician Ludolph van Ceulen, which took him decades. Soon after, more powerful methods for the computation were found.

Other implementations

Early uses of sequences of nested intervals (or can be described as such with modern mathematics), can be found in the predecessors of calculus (differentiation and integration). In computer science, sequences of nested intervals is used in algorithms for numerical computation. E.g. the bisection method can be used for calculating the roots of continuous functions. In contrast to mathematically infinite sequences, an applied computational algorithm terminates at some point, when the desired zero has been found or sufficiently well approximated.

The construction of the real numbers

In mathematical analysis, nested intervals provide one method of axiomatically introducing the real numbers as the completion of the rational numbers, being a necessity for discussing the concepts of continuity and differentiability. Historically, Isaac Newton's and Gottfried Wilhelm Leibniz's discovery of differential and integral calculus from the late 1600s has posed a huge challenge for mathematicians trying to prove their methods rigorously; despite their success in physics, engineering and other sciences. The axiomatic description of nested intervals (or an equivalent axiom) has become an important foundation for the modern understanding of calculus.

is an Archimedean ordered field, meaning the axioms of order and the Archimedean property hold.

Definition

Source:

a sequence of nested intervals, if

1. .

.

Remark

Note that some authors refer to such interval-sequences, satisfying both properties above, as shrinking nested intervals. In this case a sequence of nested intervals refers to a sequence that only satisfies property 1.

Axiom of completeness

. In formal notation this axiom guarantees, that

1. .

Theorem

.

Proof:

Notes

• , if only considering the rationals.

• The axiom is equivalent to the existence of the infimum and supremum (proof below), the convergence of Cauchy sequences and the Bolzano–Weierstrass theorem. This means that one of the four has to be introduced axiomatically, while the other three can be successively proven.

Direct consequences of the axiom

Existence of roots

By generalizing the algorithm shown above for square roots.

Existence of infimum and supremum in bounded sets

Definition

the , if

can exist. Analogously one can define the , that is bounded from below, as the greatest lower bound of that set.

Theorem

has a supremum (infimum), if it is bounded from above (below).

Proof: Without loss of generality, that has the following two properties:

1. .

and define

Note that this interval sequence is well defined and obviously a sequence of nested intervals by construction.

be the number in every interval (whose existence is guaranteed by the axiom.

, contradicting property 2 of all sequences of nested intervals.

by definition.

Remark

. In effect the two are actually equivalent, meaning that either of the two can be introduced axiomatically.

Proof:.

Further consequences

After formally defining the convergence of sequences and accumulation points of sequences, one can also prove the Bolzano–Weierstrass theorem using nested intervals. In a follow-up, the fact, that Cauchy sequences are convergent (and that all convergent sequences are Cauchy sequences) can be proven. This in turn allows for a proof of the completeness property above, showing their equivalence.

Further discussion of related aspects

over all the naturals (i.e. the set of all points common to each interval) is that it is either the empty set, a point on the number line (called a singleton), or some interval.

.

. This is given by the Archimedean property implying that the intersection has to be empty.

The situation is different for closed intervals.

. 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.

Higher dimensions

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

See also

• Bisection

• Cantor's intersection theorem

References

• .

• .

• .