Elementary Analysis: The Theory of Calculus

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Designed for students having no previous experience with rigorous proofs, this text can be used immediately after standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, as well as for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied, while many abstract ideas, such as metric spaces and ordered systems, are avoided completely. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics, and optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.

Author(s): Kenneth A. Ross
Year: 2010

Language: English
Pages: 366

Front Cover
......Page 1
Preface
......Page 6
Contents
......Page 10
1 The Set N of Natural Numbers......Page 12
2
The Set Q of Rational Numbers......Page 17
3 The Set R of Red Numbers
......Page 23
4 The Completeness Axiom
......Page 30
5 The Symbols +infinity & -infinity......Page 38
6* A Development of R
......Page 39
7 Limits of Sequences
......Page 42
8 A Discussion about Proofs......Page 48
9 Limit Theorems for Sequences
......Page 54
10 Monotone Sequences & Cauchy Sequences
......Page 65
11 Subsequences
......Page 75
12 lim sup's and lim inf's
......Page 86
13* Some Topological Concepts in Metric Spaces
......Page 90
14 Series
......Page 101
15 Alternating Series & Integral Tests
......Page 111
16* Decimal Expansions of Real Numbers
......Page 116
17 Continuous Functions
......Page 126
18 Properties of Continuous Functions
......Page 137
19 Uniform Continuity
......Page 143
20 Limits of Functions
......Page 157
21* More on Metric Spaces: Continuity
......Page 168
22* More on Metric Spaces: Connectedness
......Page 175
23 Power Series
......Page 182
24 Uniform Convergence
......Page 188
25 More on Uniform Convergence
......Page 195
26 Differentiation & Integration of Power Series
......Page 203
27* Weierstrass's Approximation Theorem
......Page 211
28 Basic Properties of the Derivative
......Page 216
29 The Mean Value Theorem
......Page 225
30* L'Hospital's Rule
......Page 233
31 Taylor's Theorem
......Page 241
32 The Riemann Integral
......Page 254
33 Properties of the Riemann Integral
......Page 264
34 Fundamental Theorem of Calculus
......Page 272
35* Riemann-Stieltjes Integrals
......Page 279
36* Improper Integrals
......Page 303
37* A Discussion of Exponents & Logarithms
......Page 310
Appendix on Set Notation
......Page 320
Selected Hints & Answers
......Page 322
References
......Page 352
Symbols Index
......Page 356
Index
......Page 358
Back Cover
......Page 366