## A Concise Approach to Mathematical AnalysisA 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 |

363 | |

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absolutely convergent an+i an+p Archimedean property Cauchy criterion Cauchy sequence closed interval Consider continuous function converges a.e. converges uniformly countable Definition denoted disjoint diverges equicontinuous Exercise exists f is continuous f uniformly finite follows Fourier series function defined function f Hence inequality integrable functions intermediate value theorem Lebesgue integral Lemma Let f lim f lim inf lim sup liman mathematical induction mean value theorem n-th neighborhood nondecreasing nonnegative notice null set open interval open set partial sums polynomial power series proof is complete Proof Let Proof Suppose prove rational number real numbers real-valued functions resp Riemann integrable sequence of functions sequence of real series converges Show that f Solution Let step function Suppose that f uniform convergence uniformly continuous upper bound Weierstrass M-test