Elementary Analysis: The Theory of Calculus

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Springer Science & Business Media, Mar 3, 1980 - Mathematics - 264 pages
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Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.
  

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Contents

1 Introduction
1
2 The Set Q of Rational Numbers
6
3 The Set R of Real Numbers
12
4 The Completeness Axiom
19
5 The Symbols +0o and 0o
27
6 A Development of R
28
7 Limits of Sequences
31
8 A Discussion about Proofs
37
4 Sequences and Series of Functions
171
24 Uniform Convergence
177
25 More on Uniform Convergence
184
26 Differentiation and Integration of Power Series
192
27 Weierstrasss Approximation Theorem
200
5 Differentiation
205
29 The Mean Value Theorem
213
30 LHospitals Rule
222

9 Limit Theorems for Sequences
43
10 Monotone Sequences and Cauchy Sequences
54
11 Subsequences
63
12 lim sups and lim infs
75
13 Some Tbpological Concepts in Metric Spaces
79
14 Series
90
15 Alternating Series and Integral Tests
100
16 Decimal Expansions of Real Numbers
105
3 Continuity
115
18 Properties of Continuous Functions
126
19 Uniform Continuity
132
20 Limits of Functions
145
Continuity
156
Connectedness
164
31 Taylors Theorem
230
6 Integration
243
33 Properties of the Riemann Integral
253
34 Fundamental Theorem of Calculus
261
35 RiemannStieltjes Integrals
268
36 Improper Integrals
292
37 A Discussion of Exponents and Logarithms
299
Appendix on Set Notation
309
Answers
311
References
341
Symbols Index
345
Index
347
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