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

The multiple space-time scale analysis has been developed to study the long time behavior of a hierarchically interacting system of stochastic differential equations. With this tool it is possible to examine the decoupled limit of this system in large space and time scales and draw conclusions for the original system. By using the multiple space-time scale analysis the long time behavior of the system is given by a Markov chain, the so-called interaction chain. Therefore, it is easier to get results: The Markov chain takes values in the same space as each "particle" of the system, so one does not have to observe countably infinitly many particles, and time takes values in a discrete set, e.g. Z \Gamma , instead of R+ . So far the particles of the examined systems take values in [0; 1] respectively R+ . In this thesis we examine the interaction chain generated by a system with particles taking values in R. As a