## Elementary Analysis: The Theory of CalculusDesigned for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals. |

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### Contents

1 Introduction | 1 |

2 The Set Q of Rational Numbers | 6 |

3 The Set R of Real Numbers | 12 |

4 The Completeness Axiom | 19 |

5 The Symbols +0o and 0o | 27 |

6 A Development of R | 28 |

7 Limits of Sequences | 31 |

8 A Discussion about Proofs | 37 |

4 Sequences and Series of Functions | 171 |

24 Uniform Convergence | 177 |

25 More on Uniform Convergence | 184 |

26 Differentiation and Integration of Power Series | 192 |

27 Weierstrasss Approximation Theorem | 200 |

5 Differentiation | 205 |

29 The Mean Value Theorem | 213 |

30 LHospitals Rule | 222 |

9 Limit Theorems for Sequences | 43 |

10 Monotone Sequences and Cauchy Sequences | 54 |

11 Subsequences | 63 |

12 lim sups and lim infs | 75 |

13 Some Tbpological Concepts in Metric Spaces | 79 |

14 Series | 90 |

15 Alternating Series and Integral Tests | 100 |

16 Decimal Expansions of Real Numbers | 105 |

3 Continuity | 115 |

18 Properties of Continuous Functions | 126 |

19 Uniform Continuity | 132 |

20 Limits of Functions | 145 |

Continuity | 156 |

Connectedness | 164 |

31 Taylors Theorem | 230 |

6 Integration | 243 |

33 Properties of the Riemann Integral | 253 |

34 Fundamental Theorem of Calculus | 261 |

35 RiemannStieltjes Integrals | 268 |

36 Improper Integrals | 292 |

37 A Discussion of Exponents and Logarithms | 299 |

Appendix on Set Notation | 309 |

Answers | 311 |

References | 341 |

345 | |

347 | |

### Common terms and phrases

algebraic apply Theorem assertion Bolzano-Weierstrass theorem bounded function Calculus Cauchy sequence closed interval Completeness Axiom Consider continuous at x0 continuous function converges uniformly Corollary decimal expansion Dedekind cuts Definition Discussion dom(f domain equal Example exists a number f is continuous f uniformly F-integrable finite follows Formal Proof function defined function f Hence Hint holds implies f(x induction infinite L'Hospital's rule Lemma Let f Let f(x LF(f lim inf lim sup lim^oo Mathematics Mean Value Theorem metric space monotonic natural number nonnegative Note obtain open interval open sets partial sums partition polynomials power series Proof Let proof of Theorem properties r e Q radius of convergence rational numbers real number Repeat Exercise Riemann integral Riemann-Stieltjes integral Root Test series converges shows that lim sinx strictly increasing subsequence subsequential limits sup{s Suppose UF(f uniformly continuous upper bound write