Real Analysis

  1. The Real Number System
    1. Construction of Real Numbers
      1. Dedekind Cuts
        1. Cauchy Sequences of Rationals
          1. Decimal Representations
          2. Field Properties
            1. Addition Axioms
              1. Associativity of Addition
                1. Commutativity of Addition
                  1. Additive Identity
                    1. Additive Inverses
                    2. Multiplication Axioms
                      1. Associativity of Multiplication
                        1. Commutativity of Multiplication
                          1. Multiplicative Identity
                            1. Multiplicative Inverses
                            2. Distributive Property
                              1. Consequences of Field Axioms
                                1. Uniqueness of Identities
                                  1. Uniqueness of Inverses
                                    1. Cancellation Laws
                                  2. Order Properties
                                    1. Order Relation Definition
                                      1. Order Axioms
                                        1. Trichotomy Property
                                          1. Transitivity of Order
                                            1. Compatibility with Addition
                                              1. Compatibility with Multiplication
                                              2. Inequalities
                                                1. Basic Inequality Properties
                                                  1. Solving Inequalities
                                                  2. Absolute Value
                                                    1. Definition and Basic Properties
                                                      1. Triangle Inequality
                                                        1. Reverse Triangle Inequality
                                                          1. Distance Interpretation
                                                        2. Completeness Property
                                                          1. Bounded Sets
                                                            1. Upper Bounds
                                                              1. Lower Bounds
                                                                1. Bounded Sets
                                                                2. Supremum and Infimum
                                                                  1. Least Upper Bound Property
                                                                    1. Greatest Lower Bound
                                                                      1. Existence and Uniqueness
                                                                        1. Characterizations
                                                                        2. Consequences of Completeness
                                                                          1. Archimedean Property
                                                                            1. Density of Rationals
                                                                              1. Density of Irrationals
                                                                                1. Nested Interval Property
                                                                              2. Extended Real Numbers
                                                                                1. Definition and Notation
                                                                                  1. Arithmetic Operations
                                                                                    1. Order Relations
                                                                                      1. Topological Considerations

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                                                                                    3. Sequences of Real Numbers