A First Course in Analysis

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Springer Science & Business Media, Mar 11, 1994 - Mathematics - 279 pages
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The first course in Analysis, which follows calculus, along with other courses, such as differential equations and elementary linear algebra, in the curricu lum, presents special pedagogical challenges. There is a change of stress from computational manipulation to "proof. " Indeed, the course can become more a course in Logic than one in Analysis. Many students, caught short by a weak command of the means of mathematical discourse and unsure of what is expected of them, what "the game" is, suffer bouts of a kind of mental paralysis. This text attempts to address these problems in several ways: First, we have attempted to define "the game" as that of "inquiry," by using a form of exposition that begins with a question and proceeds to analyze, ultimately to answer it, bringing in definitions, arguments, conjectures, exam ples, etc. , as they arise naturally in the course of a narrative discussion of the question. (The true, historical narrative is too convoluted to serve for first explanations, so no attempt at historical accuracy has been made; our narra tives are completely contrived. ) Second, we have kept the logic informal, especially in the course of preliminary speculative discussions, where common sense and plausibility tempered by mild skepticism-serve to energize the inquiry.
 

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Contents

Approximation The Real Numbers
35
2 Completeness Nested Intervals
38
3 Bounded Monotonic Sequences
40
4 Cauchy Sequences
45
5 The Real Number System
48
6 Countability
50
Appendix The Fundamental Theorem of Algebra Complex Numbers
53
The ExtremeValue Problem
62
1 Differential and Derivative Tangent Line
133
2 The Foundations of Differentiation
138
3 Curve Sketching The MeanValue Theorem
144
4 Taylors Theorem
152
5 Functions Defined Implicitly
158
Integration
167
1 Definitions Darboux Theorem
169
2 Foundations of Integral Calculus The Fundamental Theorem of Calculus
177

1 Continuity Compactness and the ExtremeValue Theorem
63
2 Continuity of Rational Functions Limits of Sequences
70
Completion of the Proof of the Fundamental Theorem of Algebra
74
3 Sequences and Series of Reals The Number e
76
4 Sets of Reals Limits of Functions
88
Continuous Functions
95
2 Inverse Functions 𝑥ʳ for 𝑟 ℚ
99
3 Continuous Extension Uniform Continuity The Exponential and Logarithm
102
4 The Elementary Functions
107
5 Uniformity The HeineBorel Theorem
110
6 Uniform Convergence A Nowhere Differentiable Continuous Function
116
7 The Weierstrass Approximation Theorem
122
Summary of the Main Properties of Continuous Functions
126
FOUNDATIONS OF CALCULUS
129
Differentiation
131
3 The Nature of Integrability Lebesgues Theorem
187
4 Improper Integral
195
5 Arclength Bounded Variation
204
A Word About the Stieltjes Integral and Measure Theory
213
Infinite Series
216
1 The Vibrating String
217
General Considerations
222
Series of Positive Terms
228
4 Computation with Series
235
5 Power Series
242
6 Fourier Series
252
Bibliography
264
Index
267
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