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1 avr 2015 · The first part of the book covers the basic machinery of real analysis, focusing For example, consider omitting Background, Series, and able to connect abstract statements with concrete examples, and having hypothesis does not follow from ZFC, so we are also free to not assume the gener-
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In modern abstract mathematics a collection of real numbers (or any other kind of mathematical objects) is called a set. Below are some examples of sets of real
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Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Exploratory Examples for Real Analysis Joanne E. Snow and Kirk E. Weller.
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Real Analysis is the formalizationof everything we learned in Calculus This enables you to make use of the examples andintuition from your calculus courses which may help you with your proofs Throughout thecourse we will be formally proving and exploring the inner workings of the Real NumberLine (hence the nameReal Analysis)
4 (a) Suppose fn: A ? R is uniformly continuous on A for every n ? N and fn ? f uniformly on A Prove that f is uniformly continuous on A (b) Does the result in (a) remain true if fn ? f pointwise instead of uni-
Real Analysis and Multivariable Calculus Igor Yanovsky 2005 6 Problem (F’01 #4) The set of all sequences whose elements are the digits 0 and 1 is not countable Let S be the set of all binary sequences We want to show that there does not exist a one-to-one mapping from the set Nonto the set S Proof
Problems and Solutions in Real and Complex Analysis Integration Functional Equations and Inequalities by Willi-Hans Steeb International School for Scienti c Computing at University of Johannesburg South Africa Preface The purpose of this book is to supply a collection of problems in analysis
FINAL EXAMINATION SOLUTIONS MAS311 REAL ANALYSIS I QUESTION 1 (a) Show that ? 3 is irrational (10 marks) Proof Suppose that ? 3 is rational and ? 3 = p/q with integers p and q not both divisible by 3 We get the relation p2 = 3q2 from which we infer that p2 is divisible by 3 Hence p itself is divisible by 3 as 3 is a prime
SAMPLE QUESTIONS FOR PRELIMINARY REAL ANALYSIS EXAM VERSION 2 0 Contents 1 Undergraduate Calculus 1 2 Limits and Continuity 2 3 Derivatives and the Mean Value Theorem 3 4 In nite Series 3 5 The Riemann Integral and the Mean Value Theorem for Integrals 4 6 Improper Integrals 5 7 Uniform Continuity; Sequences and Series of Functions 6 8
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Real Analysis and Multivariable Calculus Igor Yanovsky, 20056 Problem (F’01, #4). The set of all sequences whose elements are the digits0and1 is not countable. LetSbe the set of all binary sequences. We want to show that there does not exist a one-to-one mapping from the set Nontothe setS. Proof.
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Real Analysis and Multivariable Calculus Igor Yanovsky, 20057 2 Unions, Intersections, and Topology of Sets Theorem. Let E?be a collection of sets.
What is the proof that RX Af(t)dt is deflned by f(x)?
Ris de?ned by F(x) = Zx a f(t)dt; then F is di?erentiable and F0=f. Proof. Sincefis continuous,F(x) = Rx af(t)dtis de?ned for allx 2[a;b].
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