Contents 1. Real Numbers 3 2 3 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 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 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 of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5