## A First Course in Real AnalysisThis book is designed for a first course in real analysis following the standard course in elementary calculus. Since many students encounter rigorous mathematical theory for the first time in this course, the authors have included such elementary topics as the axioms of algebra and their immediate consequences as well as proofs of the basic theorems on limits. The pace is deliberate, and the proofs are detailed. The emphasis of the presentation is on theory, but the book also contains a full treatment (with many illustrative examples and exercises) of the standard topics in infinite series, Fourier series, multidimensional calculus, elements of metric spaces, and vector field theory. There are many exercises that enable the student to learn the techniques of proofs and the standard tools of analysis. In this second edition, improvements have been made in the exposition, and many of the proofs have been simplified. Additionally, this new edition includes an assortment of new exercises and provides answers for the odd-numbered problems. |

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### Contents

III | 1 |

IV | 9 |

V | 15 |

VI | 25 |

VII | 30 |

VIII | 35 |

IX | 42 |

X | 48 |

XLI | 230 |

XLII | 241 |

XLIII | 250 |

XLIV | 254 |

XLV | 263 |

XLVI | 270 |

XLVII | 275 |

XLVIII | 285 |

XI | 55 |

XII | 59 |

XIII | 62 |

XIV | 68 |

XV | 70 |

XVI | 72 |

XVII | 75 |

XVIII | 77 |

XIX | 83 |

XX | 94 |

XXI | 98 |

XXII | 111 |

XXIII | 117 |

XXIV | 122 |

XXV | 130 |

XXVI | 136 |

XXVII | 145 |

XXVIII | 150 |

XXIX | 157 |

XXX | 161 |

XXXI | 164 |

XXXII | 173 |

XXXIII | 178 |

XXXIV | 188 |

XXXV | 194 |

XXXVI | 197 |

XXXVII | 203 |

XXXVIII | 211 |

XXXIX | 216 |

XL | 222 |

XLIX | 290 |

L | 295 |

LI | 305 |

LII | 316 |

LIII | 329 |

LIV | 335 |

LV | 341 |

LVI | 348 |

LVII | 359 |

LVIII | 369 |

LIX | 374 |

LX | 381 |

LXI | 393 |

LXII | 403 |

LXIII | 413 |

LXIV | 423 |

LXV | 434 |

LXVI | 445 |

LXVII | 455 |

LXVIII | 461 |

LXIX | 471 |

LXX | 477 |

LXXI | 486 |

LXXII | 495 |

LXXIII | 499 |

LXXIV | 503 |

LXXV | 507 |

LXXVI | 515 |

LXXVII | 529 |

### Common terms and phrases

Axiom bounded variation calculus Cauchy sequence Chain rule closed interval compact contains continuous function converges uniformly convex function convex set coordinate system countable Darboux integrable definition denoted differentiable diverges domain equation establish example f is continuous Figure finite number follows formula Fourier series function f function theorem functions defined given Hence hypercube Implicit function theorem induction inequality infinite interior point inverse Lemma Let F limit point linear mapping metric space natural numbers nondecreasing obtain open ball open interval open set partial derivatives piecewise smooth polynomial positive integer positive number Problems proof of Theorem properties Prove Theorem rational numbers reader real numbers rectangle result Riemann integral Riemann-Stieltjes integral S+(f scalar Section Show smooth surface element solution subdivision subinterval subset Suppose that f uniform convergence uniformly continuous unique zero