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Introduction to Infinitesimal Analysis by Oswald Veblen and N Project Gutenberg’s Introduction to Infinitesimal Analysis by Oswald Veblen and N. J. Lennes This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net Title: Introduction to Infinitesimal Analysis Functions of one real variable Author: Oswald Veblen and N. J. Lennes Release Date: July 2, 2006 [EBook #18741] Language: English Character set encoding: TeX *** START OF THIS PROJECT GUTENBERG EBOOK INFINITESIMAL ANALYSIS *** Produced by K.F. Greiner, Joshua Hutchinson, Laura Wisewell, Owen Whitby and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by Cornell University Digital Collections.) 2 Transcriber’s Notes. A large number of printer errors have been corrected. These are shaded like this, and details can be found in the source code in the syntax \correction{corrected}{original}. In addition, the formatting of a few lem- mas, corollaries etc. has been made consistent with the others. = The unusual inequality sign > used a few times in the book in addition to = has been preserved, although it may reflect the printing rather than the author’s intention. The | | notation a b for intervals is not in common use today, and the reader able to run LATEX will find it easy to redefine this macro to give a modern equivalent. Similarly, the original did not mark the ends of proofs in any way and so nor does this version, but the reader who wishes can easily redefine \qedsymbol in the source. ii INTRODUCTION TO INFINITESIMAL ANALYSIS FUNCTIONS OF ONE REAL VARIABLE BY OSWALD VEBLEN Preceptor in Mathematics, Princeton University And N. J. LENNES Instructor in Mathematics in the Wendell Phillips High School, Chicago FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS London: CHAPMAN & HALL, Limited 1907 ii Copyright, 1907 by OSWALD VEBLEN and N. J. LENNES ROBERT DRUMMOND, PRINTER, NEW YORK PREFACE A course dealing with the fundamental theorems of infinitesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician. It appears in the curriculum of nearly every university, and is taken by students as “Advanced Calculus” in their last collegiate year, or as part of “Theory of Functions” in the first year of graduate work. This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources. The book may also be used as a basis for a rather short theoretical course on real functions, such as is now given from time to time in some of our universities. The general aim has been to obtain rigor of logic with a minimum of elaborate machin- ery. It is hoped that the systematic use of the Heine-Borel theorem has helped materially toward this end, since by means of this theorem it is possible to avoid almost entirely the sequential division or “pinching” process so common in discussions of this kind. The definition of a limit by means of the notion “value approached” has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of limits, and in general has largely decreased the number of ε’s and δ’s. The theory of limits is developed for multiple-valued functions, which gives certain advantages in the treatment of the definite integral. In each chapter the more abstract subjects and those which can be omitted on a first reading are placed in the concluding sections. The last chapter of the book is more advanced in character than the other chapters and is intended as an introduction to the study of a special subject. The index at the end of the book contains references to the pages where technical terms are first defined. When this work was undertaken there was no convenient source in English containing a rigorous and systematic treatment of the body of theorems usually included in even an elementary course on real functions, and it was necessary to refer to the French and German treatises. Since then one treatise, at least, has appeared in English on the Theory of Functions of Real Variables. Nevertheless it is hoped that the present volume, on account of its conciseness, will supply a real want. The authors are much indebted to Professor E. H. Moore of the University of Chicago for many helpful criticisms and suggestions; to Mr. E. B. Morrow of Princeton University for reading the manuscript and helping prepare the cuts; and to Professor G. A. Bliss of Princeton, who has suggested several desirable changes while reading the proof-sheets. iii iv Contents 1 THE SYSTEM OF REAL NUMBERS. 1 § 1 Rational and Irrational Numbers. ....................... 1 § 2 Axiom of Continuity. .............................. 2 § 3 Addition and Multiplication of Irrationals. .................. 6 § 4 General Remarks on the Number System. .................. 8 § 5 Axioms for the Real Number System. ..................... 9 § 6 The Number e. ................................. 11 § 7 Algebraic and Transcendental Numbers. ................... 14 § 8 The Transcendence of e. ............................ 14 § 9 The Transcendence of π. ............................ 18 2 SETS OF POINTS AND OF SEGMENTS. 23 § 1 Correspondence of Numbers and Points. ................... 23 § 2 Segments and Intervals. Theorem of Borel. .................. 24 § 3 Limit Points. Theorem of Weierstrass. .................... 28 § 4 Second Proof of Theorem 15. ......................... 31 3 FUNCTIONS IN GENERAL. SPECIAL CLASSES OF FUNCTIONS. 33 § 1 Definition of a Function. ............................ 33 § 2 Bounded Functions. ............................... 35 § 3 Monotonic Functions; Inverse Functions. ................... 36 § 4 Rational, Exponential, and Logarithmic Functions. ............. 41 4 THEORY OF LIMITS. 47 § 1 Definitions. Limits of Monotonic Functions. ................. 47 § 2 The Existence of Limits. ............................ 51 § 3 Application to Infinite Series. ......................... 55 § 4 Infinitesimals. Computation of Limits. .................... 58 § 5 Further Theorems on Limits. .......................... 64 § 6 Bounds of Indetermination. Oscillation. .................... 65 v vi CONTENTS 5 CONTINUOUS FUNCTIONS. 69 § 1 Continuity at a Point. ............................. 69 § 2 Continuity of a Function on an Interval. ................... 70 § 3 Functions Continuous on an Everywhere Dense Set. ............. 74 § 4 The Exponential Function. ........................... 76 6 INFINITESIMALS AND INFINITES. 81 § 1 The Order of a Function at a Point. ...................... 81 § 2 The Limit of a Quotient. ............................ 84 § 3 Indeterminate Forms .............................. 86 § 4 Rank of Infinitesimals and Infinites. ...................... 91 7 DERIVATIVES AND DIFFERENTIALS. 93 § 1 Definition and Illustration of Derivatives. ................... 93 § 2 Formulas of Differentiation. .......................... 95 § 3 Differential Notations. ............................. 102 § 4 Mean-value Theorems. ............................. 104 § 5 Taylor’s Series. ................................. 107 § 6 Indeterminate Forms. .............................. 111 § 7 General Theorems on Derivatives. ....................... 115 8 DEFINITE INTEGRALS. 121 § 1 Definition of the Definite Integral. ....................... 121 § 2 Integrability of Functions. ........................... 124 § 3 Computation of Definite Integrals. ....................... 128 § 4 Elementary Properties of Definite Integrals. ................. 132 § 5 The Definite Integral as a Function of the Limits of Integration. ...... 138 § 6 Integration by Parts and by Substitution. ................... 141 § 7 General Conditions for Integrability. ...................... 143 9 IMPROPER DEFINITE INTEGRALS. 153 § 1 The Improper Definite Integral on a Finite Interval. ............. 153 § 2 The Definite Integral on an Infinite Interval. ................. 161 § 3 Properties of the Simple Improper Definite Integral. ............. 164 § 4 A More General Improper Integral. ...................... 168 § 5 Existence of Improper Definite Integrals on a Finite Interval ........ 174 § 6 Existence of Improper Definite Integrals on the Infinite Interval ...... 178 Chapter 1 THE SYSTEM OF REAL NUMBERS. § 1 Rational and Irrational Numbers. The real number system may be classified as follows: (1) All integral numbers, both positive and negative, including zero. m (2) All numbers n , where m and n are integers (n 6= 0). √ (3) Numbers not included in either of the above classes, such as 2 and π.1 Numbers of classes (1) and (2) are called rational or commensurable numbers, while the numbers of class (3) are called irrational or incommensurable numbers. As an illustration√ of an irrational number consider the square root of 2. One ordinarily says that 2 is 1.4+, or 1.41+, or 1.414+, etc. The exact meaning of these statements is expressed by the following inequalities:2 (1.4)2 < 2 < (1.5)2, (1.41)2 < 2 < (1.42)2, (1.414)2 < 2 < (1.415)2, etc. Moreover, by the foot-note above no terminating decimal is equal to the square root of 2. Hence Horner’s Method, or the usual algorithm for extracting the square root, leads to an 1 m m2 m2 2 2 2 It is clear that there is no number n such that n2 = 2, for if n2 = 2, then m = 2n , where m and 2n2 are integral numbers, and 2n2 is the square of the integral number m. Since in the square of an integral
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