## A First Course in AnalysisThe 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 |

267 | |

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absolute convergence Algebra antiderivative approximation Archimedean ordered field Archimedean property bounded function bounded variation Calculus Cauchy sequence choose closed interval cluster point coefficients compact complete ordered field complex numbers computation consider content zero continuous function countable Darboux Theorem decimal defined Definition denoted derivative difference quotient digits diverges domain endpoints example EXERCISE 11 Exercise 9 exists expression f is continuous finite follows formula Fourier series function f Fundamental Theorem geometric gives graph hence hypothesis implies improper integrals induction inequality infinitely integrable intuition inverse length lim xn limit point Mathematics monotonic multiplication natural numbers neighborhood of x0 Nested Intervals notation number system one-one open interval open set ordered field partial sums partition polynomial power series Proof Proposition Prove rational real number Remark representation result Riemann sums Show solution Stieltjes integral subintervals subset Suppose term-by-term tion uniformly upper bound yields

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Page 3 - RC, the total sample size less the total number of cells in the design represented as the product of the number of rows by the number of columns (RC).