Introduction to Real Analysis Dr. Weihu Hong Clayton State University 8/19/2008

Review of Real Number System R denotes the set of real numbers; Q denotes the set of rational numbers; (R, +, ) forms a field; A field is a set F with two operations, addition (+) and multiplication (), that satisfy the following axioms: F is closed under both operations; Both operations are commutative; Both operations are associative; There exist additive identity 0 and multiplicative identity 1; Every element has an additive inverse; Every nonzero element has a multiplicative inverse Multiplication is distributive over addition. Is (Q, +, ) a field?

Order Properties P denotes the set of positive real numbers. The following are two order properties: (O1) If a and b belong to P, then so do a + b and ab. (O2) If a belongs to R, then one and only one of the following hold: Any field F with a nonempty subset satisfying the order properties is called an ordered field. Therefore, (R, +, ) is an ordered field. Is (Q, +, ) an ordered field? Is (C, +, ) an ordered field? Where C denotes the set of complex numbers.

The meanings of Inequality signs Let a and b belong to R. If a – b belongs to P, then we write a > b or b < a; If a – b belongs to P or a – b = 0, then we write a b. Basic results:

Upper Bound of a set Definition of upper bound of a set: A subset E of R is bounded above if there exists β in R such that for each x in E, x β. Such a β is called an upper bound of E. How should you define “bounded below” and “lower bound”? A set E is bounded if E is bounded both above and below. Examples: A = {0, ½, 2/3, ¾, …}. Is A bounded below? Is A bounded above? Is A bounded? N = {1, 2, 3, …}. Is N bounded below? Is N bounded above? Is N bounded? B = {r belongs to Q: 0 < r and r²<2}. Is B bounded below? Is B bounded above? Is B bounded? Does B have the maximum element?