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RealVarNotes - Real Variables Lectures and Problems...

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Real Variables: Lectures and Problems September 8, 2009 Steen Pedersen Department of Mathematics, Wright State University, Dayton OH 45435, USA E-mail address : [email protected]
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Introduction These notes are an evolving document. I hope that you will help me improve the presentation and the selection of topics. Claims stated without proofs as well as all claims made in exer- cises/problems are meant to be proven by the reader. If a problem contains an error that makes it trivial, then it is the responsibility of the reader to fix the problem in a manner that does not make the problem completely trivial, yet makes it solvable. Some easy results and exercises are stated because they are useful when proving some of the more interesting statements. There are three fundamental theorems in elementary mathematics. The fundamental theorem of arithmetic: every natural number is a product of primes. The fundamental theorem of algebra: every poly- nomial has a root. And the fundamental theorem of calculus: relating the area and tangent problems in geometry. All three fundamental theorems are proven in these notes. I have deliberately tried to keep the use of R to a minimum in the first part of these notes. It is my hope that this will help the student get used to working with the definitions and not depending on vague intuitive arguments. We all think we know what a real number is and we have an intuitive understanding of convergence and hence of continuity. It is common to think of a real number as an infinite decimal e.g., π = 3 . 14159 . . . . What is the meaning of an infinite decimal? We might say π 3 , π 3 . 1 , π 3 . 14 , etc. understanding that the more digits we include the more closer we are getting to π. Of course this instantly mixes up ideas of convergence with ideas of what a real number is. Now is seems difficult to talk about convergence before we know what a real number is, but we have just seen that our intuitive understanding of the real numbers makes this necessary. In these notes it is shown that the real numbers can be constructed from the counting numbers and that in turn the counting numbers can be constructed based on set theory and induction, making induction a fundamental axiom for all of analysis. This construction resolves the iii
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iv INTRODUCTION problem: what comes first number or convergence? that I mentioned above. Hence all of analysis can be derived from set theory, logic, and the axiom of induction. Hence we write down a few axioms and everything is beautiful. Or is it? David Hilbert proposed: ”The chief requirement of the theory of ax- ioms must go farther [than merely avoiding known paradoxes], namely, to show that within every field of knowledge contradictions based on the underlying axiom-system are absolutely impossible.” Kurt G¨ odel showed that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. Hence we will never know whether or not analysis is consistent or not.
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