Constructive Analysis with Witnesses

1. Real Numbers 4 1.1. Approximation of √ 2 4 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 11 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 12 2.1. Completeness 12 2.2. Limits and Inequalities 13 2.3. Series 14 2.4. Redundant Dyadic Representation of Reals 15 2.5. Convergence Tests 16 2.6. Reordering Theorem 18 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 22 4. Continuous Functions 24 4.1. Suprema and Infima 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity for Functions of More Than One Variable 32 5. Differentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5.4. Rolle’s Lemma, Mean Value Theorem 35 6. Integration 36 6.1. Riemannian sums 36 6.2. Integration and Differentiation 38 6.3. Substitution Rule, Partial Integration 40 6.4. Intermediate Value Theorem of Integral Calculus 41