Introduction To Mathematical Analysis

1 Introduction ToMathematical AnalysisJohn E. Hutchinson1994 Revised byRichard J. Loy1995/6/7 Department of MathematicsSchool of Mathematical SciencesANUPure mathematics have one peculiar advantage, that they occa-sion no disputes among wrangling disputants, as in other branchesof knowledge; and the reason is, because the de nitions of theterms are premised, and everybody that reads a proposition hasthe same idea of every part of it. Hence it is easy to put an endto all Mathematical controversies by shewing, either that our ad-versary has not stuck to his de nitions, or has not laid down truepremises, or else that he has drawn false conclusions from trueprinciples; and in case we are able to do neither of these, we mustacknowledge the truth of what he has proved:::The mathematics, he [Isaac Barrow] observes, e ectually exercise,not vainly delude, nor vexatiously torment, studious minds withobscure subtilties; but plainly demonstrate everything within theirreach, draw certain conclusions, instruct by pro table rules, andunfold pleasant questions.

2 These disciplines likewise enure andcorroborate the mind to a constant diligence in study; they whollydeliver us from credulous simplicity; most strongly fortify usagainst the vanity of scepticism, e ectually refrain us from arash presumption, most easily incline us to a due assent, per-fectly subject us to the government of right reason. While themind is abstracted and elevated from sensible matter, distinctlyviews pure forms, conceives the beauty of ideas and investigatesthe harmony of proportion; the manners themselves are sensiblycorrected and improved, the a ections composed and recti ed,the fancy calmed and settled, and the understanding raised andexited to more divine dia Britannica[1771]iPhilosophy is written in this grand book|I mean the universe|which standscontinually open to our gaze, but it cannot be understood unless one rst learnsto comprehend the language and interpret the characters in which it is is written in the language of mathematics, and its characters are triangles,circles, and other Mathematical gures, without which it is humanly impossibleto understand a single word of it.

3 Without these one is wandering about in a GalileiIl Saggiatore[1623]Mathematics is the queen of the Friedrich Gauss[1856]Thus mathematics may be de ned as the subject in which we never know whatwe are talking about, nor whether what we are saying is RussellRecent Work on the Principles of Mathematics,International Monthly, vol. 4 [1901]Mathematics takes us still further from what is human, into the region of absolutenecessity, to which not only the actual world, but every possible world, RussellThe Study of Mathematics[1902]Mathematics, rightly viewed, possesses not only truth, but supreme beauty|abeauty cold and austere, like that of a sculpture, without appeal to any partof our weaker nature, without the gorgeous trappings of painting or music, yetsublimely pure, and capable of perfection such as only the greatest art can RussellThe Study of Mathematics[1902]The study of mathematics is apt to commence in We aretoldthat by its aid the stars are weighed and the billions of molecules in a drop ofwater are counted.

4 Yet, like the ghost of Hamlet's father, this great science eludesthe e orts of our mental weapons to grasp North WhiteheadAn Introduction to Mathematics[1911]The science of pure mathematics, in its modern developments, may claim to bethe most original creation of the human North WhiteheadScience and the Modern World[1925]All the pictures which science now draws of nature and which alone seem capableof according with observational facts are Mathematical From theintrinsic evidence of his creation, the Great Architect of the Universe now beginsto appear as a pure James Hopwood JeansThe Mysterious Universe[1930]A mathematician, like a painter or a poet, is a maker of patterns. If his patternsare more permanent than theirs, it is because they are made of HardyA Mathematician's Apology[1940]The language of mathematics reveals itself unreasonably e ective in the , a wonderful gift which we neither understand nor deserve.

5 We shouldbe grateful for it and hope that it will remain valid in future research and that itwill extend, for better or for worse, to our pleasure even though perhaps to ourba ement, to wide branches of Wigner[1960]iiTo instruct someone:::is not a matter of getting him (sic) to commit results tomind. Rather, it is to teach him to participate in the process that makes possiblethe establishment of knowledge. We teach a subject not to produce little livinglibraries on that subject, but rather to get a student to think mathematically forhimself:::to take part in the knowledge getting. Knowing is a process, not BrunerTowards a theory of instruction[1966]The same pathological structures that the mathematicians invented to break loosefrom 19-th naturalism turn out to be inherent in familiar objects all around us DysonCharacterising Irregularity,Science 200 [1978]Anyone who has been in the least interested in mathematics, or has even observedother people who are interested in it, is aware that Mathematical work is workwith ideas.

6 Symbols are used as aids to thinking just as musical scores are used inaids to music. The music comes rst, the score comes later. Moreover, the scorecan never be a full embodiment of the musical thoughts of the composer. Justso, we know that a set of axioms and de nitions is an attempt to describe themain properties of a Mathematical idea. But there may always remain as aspectof the idea which we use implicitly, which we have not formalized because we havenot yet seen the counterexample that would make us aware of the possibility ofdoubting 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 byphysical objects). What are the main properties of Mathematical activity or math-ematical knowledge, as known to all of us from daily experience?

7 (1) Mathematicalobjects are invented or created by humans. (2) They are created, not arbitrarily,but arise from activity with already existing Mathematical objects, and from theneeds of science and daily life. (3) Once created, Mathematical objects have prop-erties which are well-determined, which we may have great di culty discovering,but which are possessed independently of our knowledge of HershAdvances in Mathematics31[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? Whathappens in the classical special case? What about the degenerate cases? Wheredoes the proof use the hypothesis?Paul HalmosI Want to be a Mathematician[1985]Mathematics is like a ight of fancy, but one in which the fanciful turns out tobe real and to have been present all along.

8 Doing mathematics has the feel offanciful invention, but it is really a process for sharpening our perception so thatwe discover patterns that are everywhere around.:::To share in the delight andthe intellectual experience of mathematics { to y where before we walked { thatis the goal of Mathematical feature of mathematics which requires special care:::is its \height", thatis, 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 canbe relied upon. This means that it is possible to build conceptual structures atonce very tall, very reliable, and extremely powerful. The structure is not like atree, but more like a sca olding, with many interconnecting supports. Once thesca olding is solidly in place, it is not hard to build up higher, but it is impossibleto build a layer before the previous layers are in Thurston,Notices Amer.}}

9 Math. Soc. [1990]Contents1 Preliminary History of Why \Prove" Theorems?.. \Summary and Problems" The approach to be 32 Some Elementary Mathematical Quanti Order of Quanti .. Truth Proofs .. of Statements Involving of Statements Involving \There Exists".. of Statements Involving \For Every".. by 183 The Real Number Algebraic of the Algebraic Sets of Real Order and Lower *Existence and Uniqueness of the Real Number System Archimedean 304 Set Russell's Union, Intersection and Di erence of as Associated with Properties of Equivalence of Denumerable Uncountable Cardinal More Properties of Sets of *Further The Axiom of Other Cardinal The Continuum Cardinal Ordinal 555 Vector Space Properties of Vector Normed Vector Inner Product 606 Metric Basic Metric Notions General Metric Interior, Exterior.

10 Boundary and Open and Closed Metric 737 Sequences and Convergence of Elementary Sequences Sequences and Components Sequences and the Closure of a Algebraic Properties of 848 Cauchy Cauchy Complete Metric Contraction Mapping 929 Sequences and Existence of Convergent Compact Nearest Limits of Diagrammatic Representation of De nition of Equivalent De Elementary Properties of Continuity at a Basic Consequences of Lipschitz and H older Another De nition of Continuous Functions on Compact Uniform Uniform Convergence of Discussion and De The Uniform Uniform Convergence and Uniform Convergence and Uniform Convergence and Di First Order Systems of Di erential Predator-Prey A Simple Spring Reduction to a First Order Initial Value Heuristic Justi cation for theExistence of Phase Space Examples of Non-Uniquenessand A Lipschitz Reduction to an Integral Local of Results to Koch Cantor Sierpinski Fractals and Dimension of Fractals as Fixed *The Metric Space of Compact Subsets *Random De Compactness and Sequential *Lebesgue covering Consequences of A Criterion for Equicontinuous Families of Arzela-Ascoli Peano's Existence Connected Connectedness Path Connected Basic Di erentiation of Real-Valued Algebraic Partial Directional The Di erential (or Derivative).