Unformatted text preview: if there exists a number K ∈ R such that x ≤ K for every x ∈ S . The number K is called an upper bound of the set S . COMPLETENESS AXIOM. Suppose that S is a nonempty set of real numbers and S is bounded above. Then there is a real number M ∈ R such that M ≤ K for every upper bound K of the set S , and that M > L for any real number L that is not an upper bound of S . Remark. The crucial assertion is that this number M is a real number. The set S = {x ∈ Q : x2 < 2} √ is bounded above. We can take K = 2 or K = 52000 . However, we clearly have M = 2. 1.4. Further Discussion on the Real Numbers In this optional section, we shall ﬁrst of all demonstrate the equivalence of the condition (WO) and the two forms of the Principle of induction. Proof of the equivalence of the Wellordering principle and the two Principles of induction. Our ﬁrst step is to show that the condition (WO) is equivalent to the Principle of induction (strong form) (PIS). ((WO) ⇒ (PIS)) Suppose th...
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 Fall '08
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 Math, Calculus, Real Numbers, Cos, Complex number, W W L Chen

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