A Formal Background to Mathematics 2a

VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.- VII.1.8 Theorem.- VII.1.9 Problems.- VII.1.10 Theorem.- VII.1.11 Theorem.- VII.1.12 Examples.- VII.1.13 More about converses.- VII.2 Infinite limits.- VII.2.1 The symbols -?, -?; the extended real line.- VII.2.2 Definition of convergence to ? or to -?.- VII.2.3 Theorem.- VII.2.4 Remarks.- VII.2.5 Example.- VII.2.6 Problems.- VII.3 Subsequences.- VII.3.1 Definition of subsequences.- VII.3.2 Theorem.- VII.3.3 Theorem.- VII.3.4 Examples.- VII.3.5 Lemma.- VII.3.6 Remark.- VII.4 The Monotone Convergence Principle again.- VII.4.1 The MCP.- VII.4.2 Example: the compound interest sequence.- VII.4.3 Preliminaries concering the number e.- VII.4.4 Problems.- VII.4.5 Theorem (Weierstrass-Bolzano).- VII.4.6 Kroneckers Theorem.- VII.5 Suprema and infima of sets of real numbers.- VII.5.1 Suprema.- VII.5.2 Infima.- VII.5.3 Example.- VII.5.4 Problems.- VII.5.5 Concerning formalities.- VII.5.6 Concerning notation and terminology.- VII.6 Exponential and logarithmic functions.- VII.6.1 Definition of exp.- VII.6.2 Theorem.- VII.6.3 Theorem.- VII.6.4 Remarks.- VII.6.5 Theorem.- VII.6.6 Theorem.- VII.6.7 An alternative approach.- VII.6.8 Concerning formalities.- VII.7 The General Principle of Convergence.- VII.7.1 Definition.- VII.7.2 The GCP.- VII.7.3 Discussion of convergence principles.- VII.7.4 Remarks concerning Cantors construction of R.- VII.7.5 Concerning existential proofs.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.1.1 Definition of continuous functions.- VIII.1.2 Examples.- VIII.1.3 Theorem.- VIII.1.4 Problems.- VIII.2 Properties of continuous functions.- VIII.2.1 Theorem (Intermediate Value Theorem).- VIII.2.2 Comments on the preceding proof.- VIII.2.3 Corollary.- VIII.2.4 A geometrical illustration.- VIII.2.5 Theorem.- VIII.2.6 Problems.- VIII.2.7 Theorem.- VIII.2.8 Corollary.- VIII.2.9 Remark.- VIII.2.10 Problem.- VIII.2.11 Remark.- VIII.2.12 Problems.- VIII.3 General exponential, logarithmic and power functions.- VIII.3.1 Real powers of positive numbers.- VIII.3.2 The exponential and logarithmic functions with base a.- VIII.3.3 Power functions.- VIII.3.4 Problems.- VIII.4 Limit of a function at a point.- VIII.4.1 Preliminary definitions.- VIII.4.2 The full and punctured limits of a function at a point.- VIII.4.3 Theorem.- VIII.4.4 Some formalities and further discussion.- VIII.4.5 Theorem.- VIII.4.6 Limits of composite functions.- VIII.4.7 Other species of limits; one sided limits.- VIII.4.8 Problems.- VIII.5 Uniform continuity.- VIII.5.1 Preliminary discussion.- VIII.5.2 Definition.- VIII.5.3 Theorem.- VIII.5.4 Problems.- VIII.5.5 Remarks.- VIII.6 Convergence of sequences of functions.- VIII.6.1 Definition of pointwise convergence.- VIII.6.2 Examples.- VIII.6.3 Further discussion.- VIII.6.4 Definition of uniform convergence.- VIII.6.5 Theorem.- VIII.6.6 Examples.- VIII.6.7 Theorem.- VIII.6.8 Theorem.- VIII.6.9 Discussion of some formalities.- VIII.7 Polynomial approximation.- VIII.7.1 Preliminaries.- VIII.7.2 Theorem (Weierstrass).- VIII.7.3 Theorem (Bernstein).- VIII.7.4 Remarks.- VIII.8 Another approach to expa.- Preliminaries.- VIII.8.1 Existence of a solution.- VIII.8.2 Uniqueness of the solution.- VIII.8.3 Summary.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.1.1 Definitions.- IX.1.2 Example.- IX.1.3 Theorem.- IX.1.4 Theorem.- IX.1.5 Theorem.- IX.1.6 Theorem.- IX.1.7 Examples.- IX.2 Absolute and conditional convergence.- IX.2.1 Definition of absolute and conditional convergence.- IX.2.2 Theorem.- IX.2.3 Theorem (General Comparison Test).- IX.2.4 Problems.- IX.2.5 Theorem (dAlemberts Ratio Test).- IX.2.6 Theorem (Cauchy n-th Root Test).- IX.2.7 Theorem (Leibnitz