دانلود کتاب Introduction to Mathematical Analysis

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Preface
Acknowledgments
Contents
Symbols and Notations
1 The System of Real Numbers
1.1 Peano Postulates/Axioms
1.2 The Structure of the Real Number System
1.3 Density and Completeness
1.4 Fields
1.5 Ordered Fields
1.5.1 The Absolute Value or Modulus
1.5.2 Extended Real Numbers
1.6 Neighborhoods
1.7 Functions, Countable and Uncountable Sets
1.7.1 Functions
1.7.2 Direct and Inverse Images
1.7.3 Composition of Functions
1.7.4 Equivalence of Sets
1.7.5 Countable Sets
1.7.6 Uncountable Sets
1.7.7 Bounded Above
1.7.8 Bounded Below
1.8 Bounded Sets
1.8.1 The Greatest Element (The Maximum Element)
1.8.2 The Smallest Element (The Least Element)
1.8.3 The Least Upper Bound (l.u.b.)/Supremum (Sup)
1.8.4 Properties
1.8.5 The Greatest Lower Bound (g.l.b.)/Infimum (Inf)
1.8.6 Properties
1.8.7 The Completeness Axiom
1.8.8 Complete Ordered Fields
1.9 The Archimedean Property of Real Numbers
1.10 Denseness
1.11 Exercises
2 Real Sequences
2.1 Sequences
2.1.1 The Range of Sequences
2.1.2 Equality of Sequences
2.1.3 Constant Sequences
2.2 Bounded Sequences
2.3 Neighborhoods of a Point
2.4 Convergence of the Sequences (Limit of Sequences)
2.4.1 Working Rule to Prove That
2.4.2 Increasing Sequences
2.4.3 Decreasing Sequences
2.4.4 Strictly Increasing or Decreasing
2.4.5 Monotonic Sequences
2.4.6 Monotonic Sequences and Their Convergence
2.5 Operations of Convergent Sequences
2.6 Operations of Divergent Sequences
2.7 Nested Intervals
2.8 Subsequences
2.9 Cauchy Sequences
2.9.1 Cauchy\'s General Principle for Convergence
2.10 Limits Superior and Inferior
2.11 Exercises
3 Infinite Series of Numbers
3.1 Positive Terms Series
3.1.1 Convergence and Divergence of Series
3.1.2 Fundamental Properties
3.1.3 Comparison Tests
3.1.4 Comparison Tests (Limit Theorems)
3.1.5 pp-Series Test
3.1.6 D\'Alembert\'s Ratio Test
3.1.7 Cauchy\'s Root Test
3.1.8 Raabe\'s Test
3.1.9 The Integral Test (Cauchy-Maclaurin\'s Integral Test)
3.1.10 Logarithmic Test
3.2 Alternating Series
3.3 Absolute and Conditional Convergence
3.4 Rearrangement of Terms
3.5 Exercises
4 Limits, Continuity, and Differentiability
4.1 The Limit of a Function
4.1.1 Some Theorems on Limits
4.2 Continuity
4.2.1 Sums, Products, and Quotients of Continuous Functions
4.2.2 Some Theorems on Continuous Functions
4.2.3 Compositions of Continuous Functions
4.3 The Intermediate Value Theorem
4.4 Uniform Continuity
4.5 Types of Discontinuity
4.6 Differentiability
4.7 Interpretation of the Derivative
4.8 One-Sided Derivatives
4.9 The Mean-Value Theorem
4.9.1 The Geometric Interpretation of the Mean-Value Theorem
4.10 The Consequences of the Mean-Value Problem
4.11 Taylor\'s Theorem
4.11.1 Taylor Polynomials
4.12 The L\'Hopital\'s Rule
4.13 Exercises
5 Metric Spaces
5.1 Definitions of Metric Spaces and Examples
5.1.1 The l Superscript plp-Spaces
5.1.2 Normed Linear Spaces
5.1.3 The Euclidean Space
5.1.4 The Pseudo-Metric (The Semi-Metric)
5.1.5 Quasi-metric Spaces
5.2 Limits of Sequences in left parenthesis upper X comma rho right parenthesis( X,ρ )
5.3 Limits of Functions in left parenthesis upper X comma rho right parenthesis( X,ρ )
5.3.1 The Diameter of a Set in Metric Spaces
5.3.2 The Distance Between a Point and a Set in Metric Spaces
5.3.3 The Distance Between Two Sets in Metric Spaces
5.3.4 Bounded Metric Spaces
5.4 Equivalent Metric Spaces
5.5 Product Metric Spaces
5.6 The Topology of Metric Spaces
5.6.1 Open and Closed Spheres; Balls
5.6.2 Neighborhoods
5.7 Continuity of Functions
5.8 Cluster Points
5.9 Open Sets
5.10 Closed Sets
5.11 The Closure of a Set
5.11.1 Interior Points
5.11.2 The Interior of a Set
5.11.3 Exterior Points and Exterior Sets
5.11.4 Frontier Points and Boundary Points
5.12 Subspaces of a Metric Space
5.12.1 Convergent Sequences
5.13 Cauchy Sequences
5.14 Complete Metric Spaces
5.15 Cantor Sets
5.15.1 Dense Sets
5.15.2 Separable Spaces
5.16 The Baire Category Theorem
5.17 Uniform Continuity
5.18 Homeomorphisms
5.19 The Banach Contraction Mapping Theorem
5.20 Compactness Arguments
5.20.1 The Finite Intersection Property (FIP)
5.20.2 Sequential Compactness
5.20.3 epsilonε-Net and Totally Bounded
5.20.4 Separable Metric Spaces
5.21 Connectedness
5.21.1 Separated Sets
5.22 Connected and Disconnected Sets
5.23 Exercises
6 The Riemann Integral
6.1 Partitions
6.2 The Norm or Mesh of the Partition
6.3 Tagged Partitions
6.4 Refinement (Finer)
6.5 Riemann Sums
6.6 Upper and Lower Riemann Integrals
6.7 The Riemann Integral
6.8 Properties of the Riemann Integral
6.9 The Fundamental Theorem of Calculus
6.9.1 Mean-Value and Change-of-Variable Theorems
6.10 Exercises
7 Sequences and Series of Functions
7.1 Sequences of Functions
7.2 Series of Functions
7.3 Pointwise Convergence
7.4 Uniform Convergence
7.5 Comparing Uniform with Pointwise Convergence
7.6 Uniform Convergence and Continuity
7.7 Uniform Convergence and Integration
7.8 Uniform Convergence and Differentiation
7.9 The Weierstrass Approximation Theorem
7.10 Exercises
8 The Lebesgue Integral
8.1 Outer Measure and Measurable Sets
8.1.1 Algebra and sigmaσ-Algebra
8.1.2 Outer Measure and Measurable Sets
8.1.3 Fundamental Properties for the Measure
8.2 Lebesgue Integral on double struck upper RmathbbR
8.2.1 Measurable Functions
8.2.2 Simple Functions and Integrals
8.2.3 Fundamental Properties of the Lebesgue Integral
8.2.4 The Convergence of Sequences of Functions
8.2.5 The Riemann Integral and the Lebesgue Integral
8.2.6 Riemann-Stieltjes Integral
8.3 Exercises
Selected Answers
Bibliography