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The Natural Numbers
Rational Numbers and Ordered Fields
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accumulation point addition Archimedean Assume assumption Axiom belong called Cauchy sequence characteristic Clearly closed condition connected Consider consists contains continuous contradicting converges Corollary correspondence decimal define Definition denote determine element equal Equation example EXERCISES exists fact Find finite follows function function F given Hence holds infinite intuitive isomorphic least Lemma Let F lower mathematical induction mean multiplication natural numbers negative nonempty notation Notice objects obtain one-one open interval operation ordered integral domain ordered pairs Peano system polynomial positive element positive integer prime Proof Proof Assume prove quotient rational numbers reader real numbers relation representation represented respectively result ring satisfies sentence shown stand subdomain subfield subset Theorem true unique upper bound