The Continuum: A Constructive Approach to Basic Concepts of by Rudolf Taschner

By Rudolf Taschner

In this small textual content the elemental thought of the continuum, together with the weather of metric house idea and continuity is built in the process of intuitionistic arithmetic within the feel of L.E.J. Brouwer and H. Weyl. the most beneficial properties are proofs of the recognized theorems of Brouwer about the continuity of all capabilities which are outlined on "whole" periods, the uniform continuity of all services which are outlined on compact periods, and the uniform convergence of all pointwise converging sequences of services outlined on compact durations. The positive method is attention-grabbing either in itself and as a distinction to, for instance, the formal axiomatic one.

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Extra resources for The Continuum: A Constructive Approach to Basic Concepts of Real Analysis

Example text

By definition there exists an integer m > 0 with the property [a - fJ1m > 10-mand therefore [a1m+l - [fJ1m+l > 5 x 1O- m- 1 . 28 2. Real numbers Define d = [alm+l - [Plm+l _ 10- m- 1 5 ' and define the positive integer j so big that 10- j :s 2d is guaranteed. Then for all integers n 2': j the desired inequality follows from [al n - [Pln 2': [alm+l - [Plm+l -I([alm+l - [Plm+l) - ([al n - [Pln)1 2': [alm+l - [Plm+l -I[alm+l - [al n I - I[Pln - [Plm+ll 2': [alm+l - [Plm+l - 2 x (10- m- 1 + lO-n) > ([a 1m +1 - [Plm+l - 5 x 10- m- 1) 2': 5d - 4d =d - 2 x lO-j .

4. The first assertion follows from la - PI :s P - a la - PI :s P :s the second from 5. a a - :s la - PI , la - PI . = P implies that a :s P and P :s a with the consequences a - y :s P - y :s IP - y I and y - a :s y - P :s IP - y I . 4 Convergent sequences of real numbers Thus we have P- la - y I ::: IP - y ::::: a - y ::::: 41 y I. In the same way we derive the consequences la - y I and y - P : : : y - a ::::: la - y I . Thus we have IP - y I : : : la - y I. • Point 4 of this corollary must not lead to the assumption that it would be possible to decide between the possibilities a ::::: P or P : : : a for any real numbers a, p.

To each real number e > 0 it is possible to construct a positive decimal number e < e. Dichotomy lemma. For any real numbers a, p, y the inequality a > at least one of the two inequalities a > y or y > p. Proof. e. {[a ]11+ 1 - [P]n+ 1 In > lO-n , and therefore [a]n+l - [P]n+l > lO- n - 5 x 10-n-1 =5x 10- n- 1 . 2 Order relations 37 As the numbers [a ]n+ I, [P]n+ I> 5 x lO-n - 1 are decimal numbers with exactly (n + 1) decimal places, we even have the inequality [a]n+1 - [P]n+1 ~ 6 x lO- n - 1 .

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