Chap 1 The Number System

The following two forms of the principle of induction

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Unformatted text preview: clude this possibility by stipulating that N has a least element. This is achieved by the condition (WO). It can be shown that the condition (WO) implies the Principle of induction. The following two forms of the Principle of induction are particularly useful. In fact, both are equivalent to the condition (WO), as we shall show in Section 1.4. PRINCIPLE OF INDUCTION (WEAK FORM). Suppose that the statement p(.) satisfies the following conditions: (PIW1) p(1) is true; and (PIW2) p(n + 1) is true whenever p(n) is true. Then p(n) is true for every n ∈ N. PRINCIPLE OF INDUCTION (STRONG FORM). Suppose that the statement p(.) satisfies the following conditions: (PIS1) p(1) is true; and (PIS2) p(n + 1) is true whenever p(m) is true for all m ≤ n. Then p(n) is true for every n ∈ N. Chapter 1 : The Number System page 2 of 19 First Year Calculus c W W L Chen, 1982, 2005 In the examples below, we shall illustrate some basic ideas involved in proof by induction. Example 1.2.1. We shall prove by i...
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