## Analysis: With an Introduction to ProofFor courses in Real Analysis, Advanced Calculus, and Transition to Advanced Mathematics or Proofs course. Carefully focused on reading and writing proofs, this introduction to the analysis of functions of a single real variable helps students in the transition from computationally oriented courses to abstract mathematics by its emphasis on proofs. Student oriented and instructor friendly, it features clear expositions and examples, helpful practice problems, many drawings that illustrate key ideas, and hints/answers for selected exercises. *NEW - True/False questions - (More than 250 total) located at the beginning of the exercises for each section and relating directly to the reading. *NEW - 8 new illustrations of key concepts make this the most visually compelling analysis text. *Straightforward discussion of logic - As it applies to the proving of theorems in analysis (Ch. 1). Can be covered briefly or in depth, depending on the needs of students. *Practice problems - Scattered throughout the narrative (more than 140 total). These problems relate directly to what has just been presented. Includes complete answers at the end of each section. *Fill-in-the-blank proofs. Helps stude |

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### Contents

Sets and Functions | 31 |

The Real Numbers | 87 |

Ordered Fields | 95 |

Copyright | |

22 other sections not shown

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### Common terms and phrases

accumulation point ANSWERS TO PRACTICE apply Archimedean property axiom of choice bijection cardinal number Cauchy sequence closed sets conclude consider continuous function contradiction convergent sequence converges uniformly Corollary countable DEFINITION Let denote denumerable derivative differentiable diverges element equinumerous equivalence class equivalence relation EXAMPLE Let Exercise exists a neighborhood exists a number exists a point Figure formula function defined given implies induction injective interval of convergence Justify each answer Let f(x lim sup Mark each statement mathematical mean value theorem metric space natural numbers nonempty subset obtain open cover open set ordered field ordered pairs partial sums partition positive number power series PRACTICE PROBLEMS properties radius of convergence rational numbers real number Section series converges statement True subsequential limits surjective Taylor's theorem THEOREM Let True or False unbounded uniform convergence uniformly continuous upper bound write