Proposition 2.2.14 (Strong principle of induction). Let \(m_0\) be a natural number, and let \(P(m)\) be a property pertaining to an arbitrary natural number \(m\). Suppose that for each \(m\geq m_0\), we have the following implication: if \(P(m')\) is true for all natural numbers \(m_0\leq m'<m\), then \(P(m)\) is also true. (In particular, this means that \(P(m_0)\) is true, since in this case the hypothesis is vacuous.) Then we can conclude that \(P(m)\) is true for all natrual numbers \(m\geq m_0\).
Proposition 2.2.14
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Exercise 2.2.5 (Proposition 2.2.14)
2.2.5. Prove Proposition 2.2.14. (Hint: define \(Q(n)\) to be the property that \(P(m)\) is true for all \(m_0\leq m<n\); note that \(Q(n)\) is vacuously true when \(n\leq m_0\).)
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Chapter2
Exercise 2.2.5. Prove Proposition 2.2.14.