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Elementry Real Analysis by Thomson Bruckner (Second Edition)


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Elementry Real Analysis written by Thomson Bruckner (Second Edition) . This is an other book of mathematics cover the following topics.

  • PROPERTIES OF THE REAL NUMBERS
    Introduction, The Real Number System, Algebraic Structure, Order Structure, Bounds, Sups and Infs, The Archimedean Property, Inductive Property of IN, The Rational Numbers Are Dense, The Metric Structure of R, Challenging Problems

  • SEQUENCES
    Introduction, Sequences, Sequence Examples, Countable Sets, Convergence, Divergence, Boundedness Properties of Limits, Algebra of Limits, Order Properties of Limits, Monotone Convergence Criterion, Examples of Limits, Subsequences, Cauchy Convergence Criterion, Upper and Lower Limits, Challenging Problems

  • INFINITE SUMS
    Introduction, Finite Sums, Infinite Unordered sums (Cauchy Criterion), Ordered Sums: Series (Properties, Special Series), Criteria for Convergence (Boundedness Criterion, Cauchy Criterion, Absolute Convergence), Tests for Convergence (Trivial Test, Direct Comparison Tests, Limit Comparison Tests, Ratio Comparison Test, d’Alembert’s Ratio Test, Cauchy’s Root Test, Cauchy’s Condensation Test, Integral Test, Kummer’s Tests, Raabe’s Ratio Test, Gauss’s Ratio Test, Alternating Series Test, Dirichlet’s Test, Abel’s Test), Rearrangements (Unconditional Convergence, Conditional Convergence, Comparison of ∑∞ i=1 ai and ∑i∈IN ai), Products of Series (Products of Absolutely Convergent Series, Products of Nonabsolutely Convergent Series), Summability Methods (Cesàro’s Method, Abel’s Method), More on Infinite Sums, Infinite Products, Challenging Problems

  • SETS OF REAL NUMBERS
    Introduction, Points (Interior Points, Isolated Points, Points of Accumulation, Boundary Points), Sets (Closed Sets, Open Sets), Elementary Topology, Compactness Arguments (Bolzano-Weierstrass Property, Cantor’s Intersection Property, Cousin’s Property, Heine-Borel Property, Compact Sets), Countable Sets, Challenging Problems

  • CONTINUOUS FUNCTIONS
    Introduction to Limits (Limits (ε-δ Definition), Limits (Sequential Definition), Limits (Mapping Definition), One-Sided Limits, Infinite Limits), Properties of Limits (Uniqueness of Limits, Boundedness of Limits, Algebra of Limits, Order Properties, Composition of Functions, Examples, Limits Superior and Inferior, Continuity (How to Define Continuity, Continuity at a Point, Continuity at an Arbitrary Point, Continuity on a Set), Properties of Continuous Functions, Uniform Continuity, Extremal Properties, Darboux Property, Points of Discontinuity (Types of Discontinuity, Monotonic Functions, How Many Points of Discontinuity?), Challenging Problems

  • MORE ON CONTINUOUS FUNCTIONS AND SETS
    Introduction, Dense Sets, Nowhere Dense Sets, The Baire Category Theorem (A Two-Player Game, The Baire Category Theorem, Uniform Boundedness), Cantor Sets (Construction of the Cantor Ternary Set, An Arithmetic Construction of K, The Cantor Function), Borel Sets (Sets of Type Gδ, Sets of Type Fσ), Oscillation and Continuity (Oscillation of a Function, The Set of Continuity Points), Sets of Measure Zero, Challenging Problems

  • DIFFERENTIATION
    Introduction, The Derivative (Definition of the Derivative, Differentiability and Continuity, The Derivative as a Magnification), Computations of Derivatives (Algebraic Rules, The Chain Rule, Inverse Functions, The Power Rule), Continuity of the Derivative?, Local Extrema, Mean Value Theorem (Rolle’s Theorem, Mean Value Theorem, Cauchy’s Mean Value Theorem), Monotonicity, Dini Derivates, The Darboux Property of the Derivative, Convexity, L’Hôpital’s Rule (L’Hôpital’s Rule: 0 0 Form, L’Hôpital’s Rule as x → ∞, L’Hôpital’s Rule: ∞ ∞ Form), Taylor Polynomials, Challenging Problems

  • THE INTEGRAL
    Introduction, Cauchy’s First Method (Scope of Cauchy’s First Method), Properties of the Integral, Cauchy’s Second Method, Cauchy’s Second Method (Continued), The Riemann Integral (Some Examples, Riemann’s Criteria, Lebesgue’s Criterion, What Functions Are Riemann Integrable?), Properties of the Riemann Integral, The Improper Riemann Integral, More on the Fundamental Theorem of Calculus, Challenging Problems

  • **VOLUME TWO**
  • SEQUENCES AND SERIES OF FUNCTIONS
    Introduction, Pointwise Limits, Uniform Limits (The Cauchy Criterion, Weierstrass M-Test, Abel’s Test for Uniform Convergence), Uniform Convergence and Continuity (Dini’s Theorem), Uniform Convergence and the Integral (Sequences of Continuous Functions, Sequences of Riemann Integrable Functions, Sequences of Improper Integrals), Uniform Convergence and Derivatives (Limits of Discontinuous Derivatives), Pompeiu’s Function, Continuity and Pointwise Limits, Challenging Problems

  • POWER SERIES
    Introduction, Power Series: Convergence, Uniform Convergence, Functions Represented by Power Series (Continuity of Power Series, Integration of Power Series, Differentiation of Power Series, Power Series Representations), The Taylor Series (Representing a Function by a Taylor Series, Analytic Functions), Products of Power Series (Quotients of Power Series), Composition of Power Series, Trigonometric Series (Uniform Convergence of Trigonometric Series, Fourier Series, Convergence of Fourier Series, Weierstrass Approximation Theorem)

  • THE EUCLIDEAN SPACES Rn
    The Algebraic Structure of Rn, The Metric Structure of Rn, Elementary Topology of Rn, Sequences in Rn, Functions and Mappings (Functions from Rn ? R, Functions from Rn ? Rm), Limits of Functions from Rn ? Rm (Definition, Coordinate-Wise Convergence, Algebraic Properties), Continuity of Functions from Rn to Rm, Compact Sets in Rn, Continuous Functions on Compact Sets, Additional Remarks

  • DIFFERENTIATION ON Rn
    Introduction, Partial and Directional Derivatives (Partial Derivatives, Directional Derivatives, Cross Partials), Integrals Depending on a Parameter, Differentiable Functions (Approximation by Linear Functions, Definition of Differentiability, Differentiability and Continuity, Directional Derivatives, An Example, Sufficient Conditions for Differentiability, The Differential), Chain Rules (Preliminary Discussion, Informal Proof of a Chain Rule, Notation of Chain Rules, Proofs of Chain Rules (I), Mean Value Theorem, Proofs of Chain Rules (II), Higher Derivatives), Implicit Function Theorems (One-Variable Case, Several-Variable Case, Simultaneous Equations, Inverse Function Theorem), Functions From R → Rm, Functions From Rn → Rm (Review of Differentials and Derivatives, Definition of the Derivative, Jacobians, Chain Rules, Proof of Chain Rule)

  • METRIC SPACES
    Introduction, Metric Spaces—Specific Examples, Additional Examples (Sequence Spaces, Function Spaces), Convergence, Sets in a Metric Space, Functions (Continuity, Homeomorphisms, Isometries), Separable Spaces, Complete Spaces (Completeness Proofs, Subspaces of a Complete Space, Cantor Intersection Property, Completion of a Metric Space), Contraction Maps, Applications of Contraction Maps (I), Applications of Contraction Maps (II), Systems of Equations (Example 13.79 Revisited), Infinite Systems (Example 13.80 revisited), Integral Equations (Example 13.81 revisited), Picard’s Theorem (Example 13.82 revisited), Compactness (The Bolzano-Weierstrass Property, Continuous Functions on Compact Sets, The Heine-Borel Property, Total Boundedness, Compact Sets in C[a,b], Peano’s Theorem), Baire Category Theorem (Nowhere Dense Sets, The Baire Category Theorem), Applications of the Baire Category Theorem (Functions Whose Graphs “Cross No Lines”, Nowhere Monotonic Functions, Continuous Nowhere Differentiable Functions, Cantor Sets), Challenging Problems

  • APPENDIX: BACKGROUND
    Should I Read This Chapter? Notation (Set Notation, Function Notation), What Is Analysis?, Why Proofs? Indirect Proof, Contraposition, Counterexamples, Induction, Quantifiers

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