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Unformatted text preview: m ≤ n, then p(n) is true in particular, so it follows from (PIW2) that p(n + 1) is true, and this gives (PIS2). It now follows from (PIS) that p(n) is true for every n ∈ N. ((PIW) ⇒ (PIS)) Suppose that (PIS1) and (PIS2) both hold for a statement p(.). Consider a statement q (.), where q (n) denotes the statement p(m) is true for every m ≤ n.
Chapter 1 : The Number System page 5 of 19 First Year Calculus c W W L Chen, 1982, 2005 Then the two conditions (PIS1) and (PIS2) for the statement p(.) imply respectively the two conditions (PIW1) and (PIW2) for the statement q (.). It follows from (PIW) that q (n) is true for every n ∈ N, and this clearly implies that p(n) is true for every n ∈ N. We next discuss the completeness of the real numbers in greater detail. First of all, the Completeness axiom can be stated in the following alternative way. COMPLETENESS AXIOM. Suppose that S is a non-empty set of real numbers and S is bounded above. Then there is a real number M ∈ R satisfying the following two conditions: (S1) For every x ∈ S , the inequality x ≤ M holds. (S2) For every > 0, there e...
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