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Edwards, R. E.
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A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis (Universitext)
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A Formal Background to Mathematics 2a [and] 2b: A Critical Approach to Elementary Analysis [2 volume set]
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ISBN 10: 0387905138
ISBN 13: 9780387905136
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Paperback. Condition: Near Fine. Two volume set. 23.5 x 15.5 cm. xlvi 606pp, 607- 1170pp. Parts 2a and 2b only. Yellow softcovers. Toning to spines. 2a contains chapters for Hidden hypotheses, infinite limits, subsequences, The monotone Convergence principle, exponential and logarithmic functions, General principle of Convergence, Continuity and limits of functions, Convergence of series, Differentiation, Integration, complex numbers, approximate integration, differential coefficients, lengths of curves. 2b contains chapters on Line integrals, Segmental and triangular paths, Convex sets, Standard subdivision of triangular paths, Cauchy's theorem, Cauchy's integral formula, Logarithmic functions, Complex analysis, notations, problems and solutions. Universitext. Seller Inventory # 79339
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A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis
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A Formal Background to Mathematics 2a: A Critical Approach To Elementary Analysis (Universitext)
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A Formal Background to Mathematics 2a
Published by
Springer New York Okt 1980, 1980
ISBN 10: 0387905138
ISBN 13: 9780387905136
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Taschenbuch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.- VII.1.8 Theorem.- VII.1.9 Problems.- VII.1.10 Theorem.- VII.1.11 Theorem.- VII.1.12 Examples.- VII.1.13 More about converses.- VII.2 Infinite limits.- VII.2.1 The symbols - , - ; the extended real line.- VII.2.2 Definition of convergence to or to - .- VII.2.3 Theorem.- VII.2.4 Remarks.- VII.2.5 Example.- VII.2.6 Problems.- VII.3 Subsequences.- VII.3.1 Definition of subsequences.- VII.3.2 Theorem.- VII.3.3 Theorem.- VII.3.4 Examples.- VII.3.5 Lemma.- VII.3.6 Remark.- VII.4 The Monotone Convergence Principle again.- VII.4.1 The MCP.- VII.4.2 Example: the compound interest sequence.- VII.4.3 Preliminaries concering the number e.- VII.4.4 Problems.- VII.4.5 Theorem (Weierstrass-Bolzano).- VII.4.6 Kronecker¿s Theorem.- VII.5 Suprema and infima of sets of real numbers.- VII.5.1 Suprema.- VII.5.2 Infima.- VII.5.3 Example.- VII.5.4 Problems.- VII.5.5 Concerning formalities.- VII.5.6 Concerning notation and terminology.- VII.6 Exponential and logarithmic functions.- VII.6.1 Definition of exp.- VII.6.2 Theorem.- VII.6.3 Theorem.- VII.6.4 Remarks.- VII.6.5 Theorem.- VII.6.6 Theorem.- VII.6.7 An alternative approach.- VII.6.8 Concerning formalities.- VII.7 The General Principle of Convergence.- VII.7.1 Definition.- VII.7.2 The GCP.- VII.7.3 Discussion of convergence principles.- VII.7.4 Remarks concerning Cantor¿s construction of R.- VII.7.5 Concerning existential proofs.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.1.1 Definition of continuous functions.- VIII.1.2 Examples.- VIII.1.3 Theorem.- VIII.1.4 Problems.- VIII.2 Properties of continuous functions.- VIII.2.1 Theorem (Intermediate Value Theorem).- VIII.2.2 Comments on the preceding proof.- VIII.2.3 Corollary.- VIII.2.4 A geometrical illustration.- VIII.2.5 Theorem.- VIII.2.6 Problems.- VIII.2.7 Theorem.- VIII.2.8 Corollary.- VIII.2.9 Remark.- VIII.2.10 Problem.- VIII.2.11 Remark.- VIII.2.12 Problems.- VIII.3 General exponential, logarithmic and power functions.- VIII.3.1 Real powers of positive numbers.- VIII.3.2 The exponential and logarithmic functions with base a.- VIII.3.3 Power functions.- VIII.3.4 Problems.- VIII.4 Limit of a function at a point.- VIII.4.1 Preliminary definitions.- VIII.4.2 The full and punctured limits of a function at a point.- VIII.4.3 Theorem.- VIII.4.4 Some formalities and further discussion.- VIII.4.5 Theorem.- VIII.4.6 Limits of composite functions.- VIII.4.7 Other species of limits; one sided limits.- VIII.4.8 Problems.- VIII.5 Uniform continuity.- VIII.5.1 Preliminary discussion.- VIII.5.2 Definition.- VIII.5.3 Theorem.- VIII.5.4 Problems.- VIII.5.5 Remarks.- VIII.6 Convergence of sequences of functions.- VIII.6.1 Definition of pointwise convergence.- VIII.6.2 Examples.- VIII.6.3 Further discussion.- VIII.6.4 Definition of uniform convergence.- VIII.6.5 Theorem.- VIII.6.6 Examples.- VIII.6.7 Theorem.- VIII.6.8 Theorem.- VIII.6.9 Discussion of some formalities.- VIII.7 Polynomial approximation.- VIII.7.1 Preliminaries.- VIII.7.2 Theorem (Weierstrass).- VIII.7.3 Theorem (Bernstein).- VIII.7.4 Remarks.- VIII.8 Another approach to expa.- Preliminaries.- VIII.8.1 Existence of a solution.- VIII.8.2 Uniqueness of the solution.- VIII.8.3 Summary.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.1.1 Definitions.- IX.1.2 Example.- IX.1.3 Theorem.- IX.1.4 Theorem.- IX.1.5 Theorem.- IX.1.6 Theorem.- IX.1.7 Examples.- IX.2 Absolute and conditional convergence.- IX.2.1 Definition of absolute and conditional convergence.- IX.2.2 Theorem.- IX.2.3 Theorem (General Comparison Test).- IX.2.4 Problems.- IX.2.5 Theorem (d¿Alembert¿s Ratio Test).- IX.2. Seller Inventory # 9780387905136
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A Formal Background to Mathematics 2a
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A Formal Background to Mathematics 2a
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A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis
Published by
Springer-Verlag New York Inc., 1980
ISBN 10: 0387905138
ISBN 13: 9780387905136
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A Formal Background to Mathematics 2a
Published by
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ISBN 10: 0387905138
ISBN 13: 9780387905136
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Kartoniert / Broschiert. Condition: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.-. Seller Inventory # 5911689
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A Formal Background to Mathematics 2a : A Critical Approach to Elementary Analysis
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Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.- VII.1.8 Theorem.- VII.1.9 Problems.- VII.1.10 Theorem.- VII.1.11 Theorem.- VII.1.12 Examples.- VII.1.13 More about converses.- VII.2 Infinite limits.- VII.2.1 The symbols - , - ; the extended real line.- VII.2.2 Definition of convergence to or to - .- VII.2.3 Theorem.- VII.2.4 Remarks.- VII.2.5 Example.- VII.2.6 Problems.- VII.3 Subsequences.- VII.3.1 Definition of subsequences.- VII.3.2 Theorem.- VII.3.3 Theorem.- VII.3.4 Examples.- VII.3.5 Lemma.- VII.3.6 Remark.- VII.4 The Monotone Convergence Principle again.- VII.4.1 The MCP.- VII.4.2 Example: the compound interest sequence.- VII.4.3 Preliminaries concering the number e.- VII.4.4 Problems.- VII.4.5 Theorem (Weierstrass-Bolzano).- VII.4.6 Kronecker¿s Theorem.- VII.5 Suprema and infima of sets of real numbers.- VII.5.1 Suprema.- VII.5.2 Infima.- VII.5.3 Example.- VII.5.4 Problems.- VII.5.5 Concerning formalities.- VII.5.6 Concerning notation and terminology.- VII.6 Exponential and logarithmic functions.- VII.6.1 Definition of exp.- VII.6.2 Theorem.- VII.6.3 Theorem.- VII.6.4 Remarks.- VII.6.5 Theorem.- VII.6.6 Theorem.- VII.6.7 An alternative approach.- VII.6.8 Concerning formalities.- VII.7 The General Principle of Convergence.- VII.7.1 Definition.- VII.7.2 The GCP.- VII.7.3 Discussion of convergence principles.- VII.7.4 Remarks concerning Cantor¿s construction of R.- VII.7.5 Concerning existential proofs.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.1.1 Definition of continuous functions.- VIII.1.2 Examples.- VIII.1.3 Theorem.- VIII.1.4 Problems.- VIII.2 Properties of continuous functions.- VIII.2.1 Theorem (Intermediate Value Theorem).- VIII.2.2 Comments on the preceding proof.- VIII.2.3 Corollary.- VIII.2.4 A geometrical illustration.- VIII.2.5 Theorem.- VIII.2.6 Problems.- VIII.2.7 Theorem.- VIII.2.8 Corollary.- VIII.2.9 Remark.- VIII.2.10 Problem.- VIII.2.11 Remark.- VIII.2.12 Problems.- VIII.3 General exponential, logarithmic and power functions.- VIII.3.1 Real powers of positive numbers.- VIII.3.2 The exponential and logarithmic functions with base a.- VIII.3.3 Power functions.- VIII.3.4 Problems.- VIII.4 Limit of a function at a point.- VIII.4.1 Preliminary definitions.- VIII.4.2 The full and punctured limits of a function at a point.- VIII.4.3 Theorem.- VIII.4.4 Some formalities and further discussion.- VIII.4.5 Theorem.- VIII.4.6 Limits of composite functions.- VIII.4.7 Other species of limits; one sided limits.- VIII.4.8 Problems.- VIII.5 Uniform continuity.- VIII.5.1 Preliminary discussion.- VIII.5.2 Definition.- VIII.5.3 Theorem.- VIII.5.4 Problems.- VIII.5.5 Remarks.- VIII.6 Convergence of sequences of functions.- VIII.6.1 Definition of pointwise convergence.- VIII.6.2 Examples.- VIII.6.3 Further discussion.- VIII.6.4 Definition of uniform convergence.- VIII.6.5 Theorem.- VIII.6.6 Examples.- VIII.6.7 Theorem.- VIII.6.8 Theorem.- VIII.6.9 Discussion of some formalities.- VIII.7 Polynomial approximation.- VIII.7.1 Preliminaries.- VIII.7.2 Theorem (Weierstrass).- VIII.7.3 Theorem (Bernstein).- VIII.7.4 Remarks.- VIII.8 Another approach to expa.- Preliminaries.- VIII.8.1 Existence of a solution.- VIII.8.2 Uniqueness of the solution.- VIII.8.3 Summary.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.1.1 Definitions.- IX.1.2 Example.- IX.1.3 Theorem.- IX.1.4 Theorem.- IX.1.5 Theorem.- IX.1.6 Theorem.- IX.1.7 Examples.- IX.2 Absolute and conditional convergence.- IX.2.1 Definition of absolute and conditional convergence.- IX.2.2 Theorem.- IX.2.3 Theorem (General Comparison Test).- IX.2.4 Problems.- IX.2.5 Theorem (d¿Alembert¿s Ratio Test).- IX.2.6 Theorem (Cauchy n-th Root Test).- IX.2.7 Theorem (Leibnitz¿ Test).- IX.2.8 Problem.- IX.2.9 Theorem.- IX.2.10 Problems.- IX.2.11 General remarks.- IX.3 Decimal expansions.- IX.3.1 Lemma.- IX.3.2 Lemma.- IX.3.3 Corollary.- IX.3.4 Example.- IX.3.5 Liouville numbers.- IX.4 Convergence of series of functions.- IX.4.1 Theorem.- IX.4.2 Problems.- IX.4.3 Theorem.- IX.4.4 Remark.- IX.4.5 Concluding remarks.- X: Differentiation.- and hidden hypotheses.- X.1 Derivatives.- X.1.1 Definition of derivative.- X.1.2 The derivative function.- X.1.3 Comments on the definition of derivative.- X.1.4 Equivalent formulations of X.1.1.- X.1.5 Differentiability and continuity.- X.1.6 Local nature of differentiability.- X.1.7 Derivative of jn when $$n in dot Nx$$.- X.1.8 Derivative of a constant function.- X.2 Rules for differentiation.- X.2.1 Theorem.- X.2.2 Theorem (The chain rule).- X.2.3 Theorem.- X.2.4 Derivative of jr when r is rational.- X.2.5 Derivatives of exponential, logarithmic and general power functions.- X.2.6 Implicit algebraic functions.- X.2.7 Cauchy¿s ¿singular function¿.- X.2.8 Continuous nowhere differentiable functions.- X.2.9 Concerning routine exercises.- X.3 The mean value theorem and its corollaries.- X.3.1 Mean value theorem.- X.3.2 Remarks.- X.3.3 Corollary.- X.3.4 Remarks.- X.3.5 Relations with monotonicity.- X.4 Primitives.- X.4.1 Difference of two primitives.- X.4.2 The existence problem for primitives.- X.4.3 Functions with no primitive.- X.4.4 Darboux continuity.- X.5 Higher order derivatives.- X.6 Extrema and derivatives.- X.6.1 Extremum points.- X.6.2 Local extrema.- X.6.3 Theorem.- X.6.4 Theorem.- X.6.5 Theorem.- X.6.6 Remarks.- X.6.7 Global extrema.- X.6.8 Global Extrema (continued).- X.6.9 The case of rational functions.- X.6.10 Some examples.- X.7 A differential equation and the exponential function again.- X.7.1 A conventional approach.- X.7.2 Remarks.- X.7.3 Preferred approach.- X.7.4 The exponential function refounded.- X.7.5 Proof of (10) in X.7.4.- X.7.6 Gener. 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