2: Real Numbers and Fields Last updated Save as PDF Page ID 19034 Elias Zakon University of Windsor via The Trilla Group (support by Saylor Foundation) 2.1: Axioms and Basic Definitions Real numbers can be constructed step by step: first the integers, then the rationals, and finally the irrationals. 2.2: Natural Numbers. Induction 2.2.E: Problems on Natural Numbers and Induction (Exercises) 2.3: Integers and Rationals 2.4: Upper and Lower Bounds. Completeness 2.4.E: Problems on Upper and Lower Bounds (Exercises) 2.5: Some Consequences of the Completeness Axiom 2.6: Powers with Arbitrary Real Exponents. Irrationals 2.6.E: Problems on Roots, Powers, and Irrationals (Exercises) 2.7: The Infinities. Upper and Lower Limits of Sequences 2.7.E: Problems on Upper and Lower Limits of Sequences in \(E^{*}\) (Exercises)