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Introduction to Mathematical Analysis

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Introduction To Mathematical Analysis John E. Hutchinson 1994 Revised by Richard J. Loy 1995/6/7 Department of Mathematics School of Mathematical Sciences ANU Pure mathematics have one peculiar advantage, that they occa- sion no disputes among wrangling disputants, as in other branches of knowledge; and the reason is, because the denitions of the terms are premised, and everybody that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies by shewing, either that our ad- versary has not stuck to his denitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles; and in case we are able to do neither of these, we must acknowledge the truth of what he has proved ... The mathematics, he [Isaac Barrow] observes, eectually exercise, not vainly delude, nor vexatiously torment, studious minds with obscure subtilties; but plainly demonstrate everything within their reach, draw certain conclusions, instruct by protable rules, and unfold pleasant questions. These disciplines likewise enure and corroborate the mind to a constant diligence in study; they wholly deliver us from credulous simplicity; most strongly fortify us against the vanity of scepticism, eectually refrain us from a rash presumption, most easily incline us to a due assent, per- fectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas and investigates the harmony of proportion; the manners themselves are sensibly corrected and improved, the aections composed and rectied, the fancy calmed and settled, and the understanding raised and exited to more divine contemplations. Encyclopdia Britannica [1771] i Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one rst learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other mathematical gures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth. Galileo Galilei Il Saggiatore [1623] Mathematics is the queen of the sciences. Carl Friedrich Gauss [1856] Thus mathematics may be dened as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell Recent Work on the Principles of Mathematics, International Monthly, vol. 4 [1901] Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform. Bertrand Russell The Study of Mathematics [1902] Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of perfection such as only the greatest art can show. Bertrand Russell The Study of Mathematics [1902] The study of mathematics is apt to commence in disappointment.... Weare told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the eorts of our mental weapons to grasp it. Alfred North Whitehead An Introduction to Mathematics [1911] The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Alfred North Whitehead Science and the Modern World [1925] All the pictures which science now draws of nature and which alone seem capable of according with observational facts are mathematical pictures .... From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. Sir James Hopwood Jeans The Mysterious Universe [1930] A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas. G.H. Hardy A Mathematician’s Apology [1940] The language of mathematics reveals itself unreasonably eective in the natural sciences..., a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps to our baement, to wide branches of learning. Eugene Wigner [1960] ii To instruct someone ... is not a matter of getting him (sic) to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself ... to take part in the knowledge getting. Knowing is a process, not a product. J. Bruner Towards a theory of instruction [1966] The same pathological structures that the mathematicians invented to break loose from 19-th naturalism turn out to be inherent in familiar objects all around us in nature. Freeman Dyson Characterising Irregularity, Science 200 [1978] Anyone who has been in the least interested in mathematics, or has even observed other people who are interested in it, is aware that mathematical work is work with ideas. Symbols are used as aids to thinking just as musical scores are used in aids to music. The music comes rst, the score comes later. Moreover, the score can never be a full embodiment of the musical thoughts of the composer. Just so, we know that a set of axioms and denitions is an attempt to describe the main properties of a mathematical idea. But there may always remain as aspect of the idea which we use implicitly, which we have not formalized because we have not yet seen the counterexample that would make us aware of the possibility of doubting it ... Mathematics deals with ideas. Not pencil marks or chalk marks, not physi- cal triangles or physical sets, but ideas (which may be represented or suggested by physical objects). What are the main properties of mathematical activity or math- ematical knowledge, as known to all of us from daily experience? (1) Mathematical objects are invented or created by humans. (2) They are created, not arbitrarily, but arise from activity with already existing mathematical objects, and from the needs of science and daily life. (3) Once created, mathematical objects have prop- erties which are well-determined, which we may have great diculty discovering, but which are possessed independently of our knowledge of them. Reuben Hersh Advances in Mathematics 31 [1979] Don’t just read it; ght it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Paul Halmos I Want to be a Mathematician [1985] Mathematics is like a ight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process for sharpening our perception so that we discover patterns that are everywhere around.... To share in the delight and the intellectual experience of mathematics – to y where before we walked – that is the goal of mathematical education. One feature of mathematics which requires special care ... is its “height”, that is, the extent to which concepts build on previous concepts. Reasoning in math- ematics can be very clear and certain, and, once a principle is established, it can be relied upon. This means that it is possible to build conceptual structures at once very tall, very reliable, and extremely powerful. The structure is not like a tree, but more like a scaolding, with many interconnecting supports. Once the scaolding is solidly in place, it is not hard to build up higher, but it is impossible to build a layer before the previous layers are in place. William Thurston, Notices Amer. Math. Soc. [1990] Contents 1 Introduction 1 1.1 Preliminary Remarks ....................... 1 1.2 History of Calculus ........................ 2 1.3 Why “Prove” Theorems? ..................... 2 1.4 “Summary and Problems” Book ................. 2 1.5 The approach to be used ..................... 3 1.6 Acknowledgments ......................... 3 2 Some Elementary Logic 5 2.1 Mathematical Statements .................... 5 2.2 Quantiers ............................. 7 2.3 Order of Quantiers ....................... 8 2.4 Connectives ............................ 9 2.4.1 Not ............................. 9 2.4.2 And ............................ 12 2.4.3 Or ............................. 12 2.4.4 Implies ........................... 13 2.4.5 I ............................. 14 2.5 Truth Tables ........................... 14 2.6Proofs............................... 15 2.6.1 Proofs of Statements Involving Connectives ...... 16 2.6.2 Proofs of Statements Involving “There Exists” ..... 16 2.6.3 Proofs of Statements Involving “For Every” ...... 17 2.6.4 Proof by Cases ...................... 18 3 The Real Number System 19 3.1 Introduction ............................ 19 3.2 Algebraic Axioms ......................... 19 i ii 3.2.1 Consequences of the Algebraic Axioms ......... 21 3.2.2 Important Sets of Real Numbers ............. 22 3.2.3 The Order Axioms .................... 23 3.2.4 Ordered Fields ...................... 24 3.2.5 Completeness Axiom ................... 25 3.2.6 Upper and Lower Bounds ................ 26 3.2.7 *Existence and Uniqueness of the Real Number System 29 3.2.8 The Archimedean Property ............... 30 4 Set Theory 33 4.1 Introduction ............................ 33 4.2 Russell’s Paradox ......................... 33 4.3 Union, Intersection and Dierence of Sets ...........
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