Section 5.1 127 39. (a) We prove that a set of n ~ 1 natural numbers has a smallest element. If n = 1, this is true because the single element of a set of size 1 is that set's smallest element. Now assume that k > 1 and that the statement is true for all £ in the range 1 ::; £ < k. We must prove it is true for n = Consider then a set S of k elements. Since k > 1, we can remove a single element a from S. The remaining k -1 elements have a smallest element b (by the induction hypothesis) and the smaller of a and b is the smallest element of This proves that any nonempty finite set of natural numbers has a smallest element. It then follows by Exercise 16 that any nonempty set of natural numbers has a smallest element. (b) Given a statement P involving the integer n, suppose 1. P is true for n = no, and 2. if k > no and P is true for all n in the range ::; n < k, then it's true also for We must prove P is true for all integers n ~ no. Proof. P is not true for all integers n ~ then the set S of such integers is a nonempty set
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