# A Concise Approach to Mathematical Analysis

Springer Science & Business Media, 2003 - Mathematics - 366 pages
A Concise Approach to Mathematical Analysis introduces the undergraduate student to the more abstract concepts of advanced calculus. The main aim of the book is to smooth the transition from the problem-solving approach of standard calculus to the more rigorous approach of proof-writing and a deeper understanding of mathematical analysis. The first half of the textbook deals with the basic foundation of analysis on the real line; the second half introduces more abstract notions in mathematical analysis. Each topic begins with a brief introduction followed by detailed examples. A selection of exercises, ranging from the routine to the more challenging, then gives students the opportunity to practise writing proofs. The book is designed to be accessible to students with appropriate backgrounds from standard calculus courses but with limited or no previous experience in rigorous proofs. It is written primarily for advanced students of mathematics - in the 3rd or 4th year of their degree - who wish to specialise in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques.

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### Contents

 Numbers and Functions 1 12 Subsets of ℝ 15 13 Variables and Functions 25 Sequences 35 22 Convergence and Limits 39 23 Subsequences 50 24 Upper and Lower Limits 53 25 Cauchy Criterion 57
 73 Power Series 194 74 Taylor Series 200 Local Structure on the Real Line 213 82 Neighborhoods and Interior Points 221 83 Closure Point and Closure 226 84 Completeness and Compactness 230 Continuous Functions 241 92 Functions Continuous on a Compact Set 245

 Series 65 32 Conditional Convergence 70 33 Comparison Tests Term Comparison Test 78 34 Root and Ratio Tests 82 35 Further Tests 85 Limits and Continuity 95 42 Continuity of Functions 104 43 Properties of Continuous Functions 108 44 Uniform Continuity 112 Differentiation 123 52 Mean Value Theorem 129 53 LHôspitals Rule 132 54 Inverse Function Theorems 134 55 Taylors Theorem 137 Elements of Integration 145 62 Riemann Integral 153 63 Functions of Bounded Variation 164 64 RiemannStieltjes Integral 168 Sequences and Series of Functions 177 72 Series of Functions 186
 93 StoneWeierstrass Theorem 251 94 Fixedpoint Theorem 257 95 AscoliArzelā Theorem 259 Introduction to the Lebesgue Integral 271 102 Lebesgue Integral 283 103 Improper Integral 296 104 Important Inequalities 300 Elements of Fourier Analysis 313 112 Convergent Trigonometric Series 319 113 Convergence in 2mean 322 114 Pointwise Convergence 328 Appendix 339 A2 Set Notations 341 A3 Cantors Ternary Set 343 A4 Bernsteins Approximation Theorem 344 Hints for Selected Exercises 347 Bibliography 361 Index 363 Copyright