MATH 4389 Study Guide - Final Guide: Archimedean Property, Rational Number, Irrational Number
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19 Apr 2017
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This material assumes that you are already familiar with the real number system and the represen- tation of the real numbers as points on the real line. Let n denote the set of natural numbers (positive integers). If s is a nonempty subset of n, Axiom: element m s such that m n for all n s. Note: a set which has the property that each non-empty subset has a least element is said to be well-ordered. Thus, the axiom tells us that the natural numbers are well-ordered. then s has a least element. Let s be a subset of n. if s has the following properties: 1 s, and, k s implies k + 1 s, then s = n. Proof: suppose s 6= n. let t = n s. then t 6= . Let m be the least element in t . Therefore, m 1 s which implies that (m 1) + 1 = m s, a contradiction.