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Real Numbers and Functions Calculus can be described as the study of how one quantity is affected by another, focusing on relationships that are smooth rather than erratic. This chapter sets up the basic language for describing quantities and the relationships between them. Quantities are represented by numbers and you would have seen different ...
the real numbers, and an equally thorough treatment of logarithmic, exponential and trigonometric functions. Additionally, the book contains some historical information
The Real Number System. Preliminaries. Given the fundamental importance of the real numbers in mathematics, it is important for mathematicians to have a logically sound description of the real number system.
Several properties of the real numbers will be developed from the axioms of this chapter and each of these results will play an important role in subsequent chapters in providing a rigorous treatment of the calculus of functions of one real variable.
the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration.
Properties of Real Numbers1 Theorem: For an arbitrary real number x, there is ex-actly one interger n which satisfies the inequalities n ≤ x < n+1. Proof: 1. Define: S = {m|m ∈ N,m ≤ x} 2. S is non empty b/c for any real number x there exists m ∈ N such that m < x. (I 3.12 2. pg 28) 3. S is bounded by x ⇒ supS exist. (A.10 pg. 25) 4.
The real numbers: R = {numbers on the number-line} require some real analysis for a “proper” definition. We’ll sidestep the analysis, relying instead on our less precise notions of continuity from calculus. Notice that the real numbers are ordered (from left to right) and come in three types: = −. ∪ {0} ∪ R. +. where. R+ = {positive real numbers}
