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Properties of the Real Numbers | |

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Introduction | |

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The Real Number System | |

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Algebraic Structure | |

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Order Structure | |

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Bounds | |

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Sups and Infs | |

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The Archimedean Property | |

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Inductive Property of IN | |

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The Rational Numbers Are Dense | |

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The Metric Structure of R. Challenging Problems for Chapter 1 | |

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Sequences | |

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Introduction | |

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Sequences | |

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Countable Sets | |

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Convergence | |

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Divergence | |

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Boundedness Properties of Limits | |

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Algebra of Limits | |

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Order Properties of Limits | |

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Monotone Convergence Criterion | |

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Examples of Limits | |

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Subsequences | |

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Cauchy Convergence Criterion | |

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Upper and Lower Limits | |

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Challenging Problems for Chapter 2 | |

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Infinite Sums | |

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Introduction | |

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Finite Sums | |

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Infinite Unordered Sums | |

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Ordered Sums: Series | |

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Criteria for Convergence | |

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Tests for Convergence | |

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Rearrangements | |

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Products of Series | |

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Summability Methods | |

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More on Infinite Sums | |

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Infinite Products | |

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Challenging Problems for Chapter 3 | |

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Sets of Real Numbers | |

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Introduction | |

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Points | |

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Sets | |

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Elementary Topology | |

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Compactness Arguments | |

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Countable Sets | |

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Challenging Problems for Chapter 4 | |

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Continuous Functions | |

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Introduction to Limits | |

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Properties of Limits | |

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Limits Superior and Inferior | |

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Continuity | |

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Properties of Continuous Functions | |

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Uniform Continuity | |

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Extremal Properties | |

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Darboux Property | |

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Points of Discontinuity | |

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Challenging Problems for Chapter 5 | |

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More on Continuous Functions and Sets | |

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Introduction | |

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Dense Sets | |

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Nowhere Dense Sets | |

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The Baire Category Theorem | |

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Cantor Sets | |

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Borel Sets | |

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Oscillation and Continuity | |

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Sets of Measure Zero | |

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Challenging Problems for Chapter 6 | |

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Differentiation | |

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Introduction | |

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The Derivative | |

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Computations of Derivatives | |

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Continuity of the Derivative? Local Extrema | |

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Mean Value Theorem | |

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Monotonicity | |

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Dini Derivatives | |

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The Darboux Property of the Derivative | |

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Convexity | |

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L'Hopital's Rule | |

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Taylor Polynomials | |

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Challenging Problems for Chapter 7 | |

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The Integral | |

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Introduction | |

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Cauchy's First Method | |

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Properties of the Integral | |

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Cauchy's Second Method | |

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Cauchy's Second Method (Continued) | |

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The Riemann Integral | |

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Properties of the Riemann Integral | |

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The Improper Riemann Integral | |

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More on the Fundamental Theorem of Calculus | |

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Challenging Problems for Chapter 8 | |

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Sequences and Series of Functions | |

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Introduction | |

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Pointwise Limits | |

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Uniform Limits | |

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Uniform Convergence and Continuity | |

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Uniform Convergence and the Integral | |

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Uniform Convergence and Derivatives | |

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Pompeiu's Function | |

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Continuity and Pointwise Limits | |

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Challenging Problems for Chapter 9 | |

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Power Series | |

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Introduction | |

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Power Series: Convergence | |

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Uniform Covergence | |

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Functions Represented by Power Series | |

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The Taylor Series | |

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Products of Power Series | |

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Composition of Power Series | |

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Trigonometric Series | |

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The Euclidean Spaces Rn | |

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The Algebraic Structure of Rn | |

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The Metric Structure of Rn | |

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Elementary Topology of Rn | |

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Sequences in Rn | |

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Functions and Mappings | |

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Limits of Functions from Rn to Rm | |

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Continuity of Functions from Rn to Rm | |

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Compact Sets in Rn | |

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Continuous Functions on Compact Sets | |

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Additional Remarks | |

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Differentiation on Rn | |

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Introduction | |

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Partial and Directional Derivatives | |

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Integrals Depending on a Parameter | |

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Differentiable Functions | |

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Chain Rules | |

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Implicit Function Theorems | |

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Functions from R to Rm | |

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Functions from Rn to Rm | |

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Metric Spaces | |

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Introduction | |

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Metric SpacesSpecific Examples | |

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Convergence | |

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Sets in a Metric Space | |

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Functions | |

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Separable Spaces | |

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Complete Spaces | |

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Contraction Maps | |

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Applications of Contraction Maps (I) | |

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Applications of Contraction Maps (II) | |

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Compactness | |

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Baire Category Theorem | |

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Applications of the Baire Category Theorem | |

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Challenging Problems for Chapter 13 | |

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Backgroun<$$$> | |