Nested intervals

The Intervallschachtelungsprinzip is specially used in analysis in evidence and forms in numerical mathematics the basis for some solution process.

The principle is the following: You start with a limited interval and selects from this interval a closed interval that lies entirely in the previous interval, a closed interval selects out there again and so on. If the lengths of the intervals arbitrarily small, so its length converges to zero, then there exists a real number that is contained in all intervals. Because of this property of intervals can be used to construct them with the real numbers as a number field extension of the rational numbers.

Basic ideas in the form of the argument of the complete division are already with Zeno of Elea and Aristotle.

Definition

Be rational or real number sequences, monotone increasing and monotone decreasing for all, and make the differences a null sequence, ie

Then the sequence or the intervals is called a nested intervals.

Construction of the real numbers

It is now that there is at most a rational number for each of intervals of rational numbers, which is contained in all intervals, ie satisfies for all.

But it is not true that each of intervals of rational numbers contains at least one rational number; to obtain such a property, you have to expand the set of rational numbers to the set of real numbers. This can for example be carried out with the help of nested intervals. Says one, each of intervals defining a well-defined real number, ie.

The equality of real numbers then we define the appropriate nested intervals: iff and always.

In an analogous way, the links of real numbers can be defined as links of nested intervals; For example, the sum of two real numbers as

Defined.

This so- defined system now has the desired properties, in particular is now that any of intervals of real numbers contains exactly one real number.

But of intervals are not the only way to construct the real numbers; particular, the implementation is more widely used as an equivalence class of Cauchy sequences. Furthermore, there is still the method of Dedekind cuts.

Other applications

• The intermediate value theorem of Bolzano can be demonstrated with the Intervallschachtelungsprinzip, from him, the Brouwer fixed point theorem is derived.
• The bisection is a numerical method based on the nested intervals.