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Nonstandard Analysis Simplified NONSTANDARD ANALYSIS - A SIMPLIFIED APPROACH - ROBERT A. HERRMANN arXiv:math/0310351v6 [math.GM] 5 Oct 2010 1 Nonstandard Analysis Simplified Copyright c 2003 by Robert A. Herrmann 2 Nonstandard Analysis Simplified CONTENTS Chapter 1 Filters Ultrafilters, Cofinite Filter, Principle Ultrafilters, Free Ultrafilters ................................. 6. Chapter 2 A Simple Nonstandard Model for Analysis Equivalence Classes of Sequences, Totally Ordered Field of Equivalence Classes, The Hyper-extension of Sets and Relations, The Standard Object Generator .................................... 9. Chapter 3 Hyper-set Algebra Infinite and Infinitesimal Numbers The Behavior of the Hyper and Standard Object Generators; Infinitesimals µ(0), The Infinite and Finite Numbers, *-Transform Process, Maximum Ideal µ(0)................ 15. Chapter 4 Basic Sequential Convergence Bounded Sequences, Convergent Sequences, Accumulation Point, Subsequences, Cauchy Criterion, Nonstandard Characteristics and Examples ....................... 23 Chapter 5 Advanced Sequential Convergence Double Sequences, Iterated Limits, Upper and Lower Limits, Nonstandard Characteristics and Examples ................................... 28. Chapter 6 Basic Infinite Series Hyperfinite Summation, Standard Results, Nonstandard Characteristics and Examples ...................... 33. Chapter 7 An Advance Infinite Series Concept Multiplying of Infinite Series, Nonstandard Characteristics and Examples ................... 37. Chapter 8 Additional Real Number Properties Interior, Closure, Cluster, Accumulation, Isolated Points, Boundedness, Compactness, Nonstandard Characteristics ............................ 41. 3 Nonstandard Analysis Simplified Chapter 9 Basic Continuous Function Concepts All Notions Generalized to Cluster Points, One-sided Limits and Continuity, Sum, Product and Composition of Continuous Functions, Extreme and Intermediate Value Theorems, Nonstandard Characteristics ................ 46. Chapter 10 Slightly Advanced Continuous Function Concepts Non- standard Analysis and Bolzano’s Product Theorem, Inverse Images of Open Sets, Additive Functions, Uniform Continuity, Extensions of Continuous Functions .............................. 51. Chapter 11 Basic Derivative Concepts Nonstandard Characteristics for Finite and Infinite Derivatives at Cluster Points, The Infinitesimal Differential, The Fundamental Theorem of Differentials, Order Ideals, Basic Theorems, Generalized Mean Value Theorem, L’Hospital’s Rule .................................... 55. Chapter 12 Some Advanced Derivative Concepts Nonstandard Analysis and the nth-Order Increments, nth-Order Ideals, Continuous Differentiability, Uniformly Differentiable, the Darboux Property, and Inverse Function Theorems ............................ 61. Chapter 13 Riemann Integration The Simple Partition, Fine Partitions, Upper and Lower Sums, Upper and Lower Hyperfinite Sums, The Simple Integral, The Equivalence of The Simple Integral and The Riemann Integral, The Basic Integral Theorems and How The Generalized and Lebesgue Integral Relate to Fine Partitions. .................................. 68 . Chapter 14 What Does the Integral Measure? Additive Functions and The Rectangular Property ............................. 75. Chapter 15 Generalizations Metric and Normed Linear Spaces ................. 78. Appendix Existence of Free Ultrafilters, Proof of *-Transform Process ................................... 80. References . 82. 4 Nonstandard Analysis Simplified Although the material in this book is copyrighted by the author, it may be repro- duced in whole or in part by any method, without the payment of any fees, as long as proper credit and identification is given to the author of the material reproduced. Disclaimer The material in this monograph has not been independently edited nor indepen- dently verified for correct content. It may contain various typographical or content errors. Corrections will be made only if such errors are significant. A few results, due to the simplicity of this approach, may not appear to be convincingly estab- lished. However, a change in our language or in our structure would remove any doubts that the results can be established rigorously. 5 Nonstandard Analysis Simplified 1. FILTERS For over three hundred years, a basic question about the calculus remained unanswered. Do the infinitesimals, as conceptional understood by Leibniz and Newton, exist as formal mathematical objects? This question was answer affirmatively by Robinson (1961) and the subject termed “Non- standard Analysis” (Robinson, 1966) was introduced to the scientific world. As part of this book, the mathematical existence of the infinitesimals is established and their properties investigated and applied to basic real analysis notions. I intend to write this “book” informally and it’s certainly about time that technical books be presented in a more “friendly” style. What I’m not going to do is to present an introduction filled with various historical facts and self-serving statements; statements that indicate what an enormous advancement in mathematics has been achieved by the use of Nonstandard Analysis. Rather, let’s proceed directly to this simplified approach, an approach that’s correct but an approach that cannot be used to analysis certain areas of mathematics that are not classified as elementary in character. These areas can be analysis but it requires one to consider additional specialized mathematical objects. Such specialized objects need only be considered after an individual becomes accustomed to the basic methods used within this simple approach. There are numerous exciting and thrilling new concepts and results that cannot be presented using the simple approach discussed. The “internal” objects, objects that “bound” sets that represent “concurrent” relations and saturated models are for your future consideration. The main goal is to present some of the basic nonstandard results that can be obtained without investigating such specialized objects. I’ll present a complete “Proof” for a stated result. However, one only needs to have confidence that the stated “Theorem” has been acceptably established. Indeed, if you simply are interested in how these results parallel the original notions of the “infinitesimal” and the like, you need not bother to read the proofs at all. There’s an immediate need for a few set-theoretic notions. We let IN be the set of all natu- ral numbers, which includes the zero as the first one. I assume that you understand some basic set-theoretic notation. Further, throughout this first chapter, X will always denote a nonempty set. Recall that for a given set X the set of all subsets of X exists and is called the power set. It’s usually denoted by the symbol (X). For example, let X = 0, 1, 2 . The power P { } set of X contains 8 sets. In particular, (X) = , X, 0 , 1 , 2 , 0, 1 , 0, 2 , 1, 2 , where P {∅ { } { } { } { } { } { }} ∅ denotes the empty set, which can be thought of as a set which contains “no members.” Definition 1.1. (The Filter.) (The symbol means “subset” and includes the possible ⊂ equality of sets.) A (nonempty) = (X) is called a (proper) filter on X if and only if ∅ 6 F⊂P (i) for each A, B , A B ; ∈F ∩ ∈F (ii) if A B X and A , then B ; ⊂ ⊂ ∈F ∈F (iii) / . ∅ ∈F Example 1.2. (i) Let = A X. Then [A] is the set of all subsets of X that contain A, or, ∅ 6 ⊂ ↑ more formally, [A] = x x X and A x , is a filter on X called the principal filter. ↑ { | ⊂ ⊂ } How to properly define what one means by a “finite” set has a long history. But as Suppes states “The common sense notion is that a set is finite just when it has “m” members for some non-negative integer m. [You can use our set IN to get such an “m.”] This common sense idea is technically sound . .” (1960, p. 98). The notion of finite can also be related to constants 6 Nonstandard Analysis Simplified that name members of a set and the formal expression that characterizes such a set in terms of the symbols “=” and “ ” (i.e.“or”). Notice that the empty set is a “finite set.” This association to the ∨ natural numbers is denoted by subscripts that actually represent the range values of a function. I also assume certain elementary properties of the finite sets and those sets that are not finite. For X, let = (X) have the finite intersection property. This means that each B is ∅ 6 B⊂P i ∈ B nonempty and that the “intersection” of all of the members of any other finite subset of is not B the empty set. Given such a = (X), then we can generate the “smallest” filter that contains . B 6 P B This is done by first letting ′ be the set of all subsets of (X) formed by taking the intersection B P of each nonempty finite subset of , where the intersection of the members of a set that contains B but one member is that one member. Let’s consider some basic mathematical abbreviations. The formal symbol means “and” and the formal “quantifier” means “there exists such and such.” ∧ ∃ Now take a member B of ′ and build a set composed of all subsets of X that contain B. Now do B this for all members of ′ and gather them together in a set to get the set . Hence, is the B hBi hBi set of all subsets x of X such that there exists some set B such that B is a member of ′ and B is B a subset of x. Formally, = x (x X) ( B((B ′) (B x)) . I have “forced” to hBi { | ⊂ ∧ ∃ ∈ B ∧ ⊂ } hBi contain and to have the necessary properties that makes it a filter. B There exists a very significant “filter” on infinite X defined by the notion of not being finite. C This object, once we have shown that it is a filter on X, is called the cofinite filter. [Cofinite means that the relative complement is a finite set]. Definition 1.3. (QED and iff.) Let = x (x X) (X x) is finite) . Also I will C { | ⊂ ∧ − } use the symbol for the statement “QED,” which indicates the end of the proof. Then “iff” is an abbreviation for the phrase “if and only if.” Theorem 1.4. The set is a filter on infinite X and, the intersection of all members of , C C F F = . { | ∈ C} ∅ Proof. Since X = , then there is some a X. Further, X (X a )= a implies that = . T 6 ∅ ∈ − −{ } { } C 6 ∅ So, assume that A, B . Then since X (A B) = (X A) (X B), X (A B) is a finite ∈ C − ∩ − ∪ − − ∩ subset of X.
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