Interval nesting

The interval nesting principle is used especially in analysis in proofs and forms the basis for some solution methods in numerical mathematics .

The principle is as follows: You start with a restricted interval and choose a closed interval from this interval that is completely within the previous interval, select a closed interval there again, and so on. If the lengths of the intervals become arbitrarily small, i.e. their length converges to zero, there is exactly one real number that is contained in all intervals. Because of this property, interval nesting can be used to construct the real numbers as a number range extension of the rational numbers.

Basic ideas in the form of the argument of complete division can already be found in Zeno of Elea and Aristotle .

definition

the first 4 links of an interval nesting

Let be rational or real number sequences , monotonically increasing and monotonically decreasing, for all , and the differences form a zero sequence, that is ${\displaystyle (a_{n}),(b_{n})}$ ${\displaystyle (a_{n})\;}$ ${\displaystyle (b_{n})\;}$ ${\displaystyle a_{n}\leq b_{n}\;}$ ${\displaystyle n\in \mathbb {N} \;}$ ${\displaystyle d_{n}=b_{n}-a_{n}}$

${\displaystyle \lim _{n\to \infty }(b_{n}-a_{n})=0\;}$,

then the sequence or also the intervals is called interval nesting. ${\displaystyle (J_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \left(a_{n}|b_{n}\right)_{n\in \mathbb {N} }}$ ${\displaystyle J_{n}:=[a_{n},b_{n}]}$

Construction of the real numbers

It is now true that for every interval nesting of rational numbers there is at most one rational number which is contained in all intervals, which therefore satisfies for all . ${\displaystyle s}$ ${\displaystyle a_{n}\leq s\leq b_{n}}$ ${\displaystyle n\in \mathbb {N} }$

But it is not true that every interval nesting of rational numbers contains at least one rational number ; in order to obtain such a property, one has to expand the set of rational numbers to the set of real numbers . This can be done, for example, with the help of the interval nesting. These are said to each of intervals defining a well-defined real number, so . Since intervals amounts are for illustration of the cut can all intervals of the nesting also be written: . ${\displaystyle s}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \sigma :=(J_{n})}$ ${\displaystyle \bigcap _{n\in \mathbb {N} }J_{n}=\{\sigma \in \mathbb {R} \}}$

The equality of real numbers is then defined using the corresponding nesting intervals: if and only if always and . ${\displaystyle \left(a_{n}|b_{n}\right)=\left(a'_{n}|b'_{n}\right)}$ ${\displaystyle a_{n}\leq b'_{n}}$ ${\displaystyle a'_{n}\leq b_{n}}$

In an analogous way, the links between real numbers can be defined as links between nesting intervals; for example, the sum of two real numbers is than

${\displaystyle \left(a_{n}|b_{n}\right)+\left(a'_{n}|b'_{n}\right)=\left(a_{n}+a'_{n}|b_{n}+b'_{n}\right)}$

Are defined.

This system defined in this way now has the desired properties. In particular, it is now the case that every arbitrary interval nesting of rational numbers contains exactly one real number.

Interval nesting is not the only way to construct real numbers; in particular, the construction is more widespread as an equivalence class of Cauchy sequences . There is also the method of Dedekind cuts .

Convergence of the limit consequences of an interval nesting

Let be an interval nesting that defines the number . Then ${\displaystyle ([a_{n},b_{n}])}$ ${\displaystyle \sigma }$

${\displaystyle \lim _{n\to \infty }(a_{n})=\sigma =\lim _{n\to \infty }(b_{n})}$

Proof : Let an arbitrary real one be given. To prove the convergence of the border sequences is to show that by choosing a suitable for both interval boundaries in an environment of lie. ${\displaystyle \varepsilon >0}$ ${\displaystyle (a_{n}),(b_{n})}$ ${\displaystyle n_{0}}$ ${\displaystyle n>n_{0}}$ ${\displaystyle a_{n},b_{n}}$ ${\displaystyle \varepsilon }$ ${\displaystyle \sigma }$

Since interval nesting, and therefore , is a null sequence, there exists such that for all . ${\displaystyle ([a_{n},b_{n}])}$ ${\displaystyle (d_{n})}$ ${\displaystyle d_{n}=b_{n}-a_{n}\geq 0}$ ${\displaystyle n_{0}}$ ${\displaystyle d_{n}<\varepsilon }$ ${\displaystyle n>n_{0}}$

Figuratively : For all of them , the diameter of the nesting intervals is so small that none of the interval boundaries reaches a boundary of the environment of if the considered interval should contain. ${\displaystyle n>n_{0}}$ ${\displaystyle a_{n},b_{n}}$ ${\displaystyle \varepsilon }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Bill : With is . For is with : ${\displaystyle \sigma \in [a_{n},b_{n}]}$ ${\displaystyle a_{n}\leq \sigma \leq b_{n}}$ ${\displaystyle n>n_{0}}$ ${\displaystyle 0\leq d_{n}<\varepsilon \Leftrightarrow 0\geq -d_{n}>-\varepsilon }$

• ${\displaystyle b_{n}=a_{n}+d_{n}\leq \sigma +d_{n}<\sigma +\varepsilon }$, because of is total ; ${\displaystyle \sigma -\varepsilon <\sigma \leq b_{n}}$ ${\displaystyle b_{n}\in U_{\varepsilon }(\sigma )}$
• ${\displaystyle a_{n}=b_{n}-d_{n}\geq \sigma -d_{n}>\sigma -\varepsilon }$, because of is total , q. e. d. ${\displaystyle \sigma +\varepsilon >\sigma \geq a_{n}}$ ${\displaystyle a_{n}\in U_{\varepsilon }(\sigma )}$

Other uses

• The intermediate value theorem of Bolzano can be proved with the Intervallschachtelungsprinzip.
• The bisection is a numerical method based on interval nesting.

Web links

Wikibooks: Math for Non-Freaks: Interval Nesting  - Learning and Teaching Materials

Wikibooks: Math for Non-Freaks: Interval Nesting with Rational Accuracy  - Learning and Teaching Materials

Individual evidence

1. ↑ Konrad Knopp. Theory and Application of Infinite Series . 5th edition, Springer Verlag 1964, ISBN 3-540-03138-3 .
2. ↑ Konrad Knopp. ibid , p. 21, definition 11.
3. ↑ Konrad Knopp. ibid, p. 22, sentence 12.
4. ↑ Konrad Knopp. ibid, p. 27, definition 13.
5. ↑ Konrad Knopp. ibid, p. 29, definition 14B.
6. ↑ Konrad Knopp. ibid, p. 31, definition 16.
7. ↑ Konrad Knopp. ibid, p. 41, sentence 4.