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Unformatted text preview: LECTURE NOTES ON MATHEMATICAL INDUCTION PETE L. CLARK Contents 1. Introduction 1 2. The (Pedagogically) First Induction Proof 4 3. The (Historically) First(?) Induction Proof 5 4. Closed Form Identities 6 5. More on Power Sums 7 6. Inequalities 10 7. Extending binary properties to n-ary properties 11 8. Miscellany 13 9. The Principle of Strong/Complete Induction 14 10. Solving Homogeneous Linear Recurrences 16 11. The Well-Ordering Principle 20 12. Upward-Downward Induction 21 13. The Fundamental Theorem of Arithmetic 23 13.1. Euclid’s Lemma and the Fundamental Theorem of Arithmetic 23 13.2. Rogers’ Inductive Proof of Euclid’s Lemma 24 13.3. The Lindemann-Zermelo Inductive Proof of FTA 25 References 25 1. Introduction Principle of Mathematical Induction for sets Let S be a subset of the positive integers. Suppose that: (i) 1 ∈ S , and (ii) ∀ n ∈ Z + ,n ∈ S = ⇒ n + 1 ∈ S . Then S = Z + . The intuitive justification is as follows: by (i), we know that 1 ∈ S . Now ap- ply (ii) with n = 1: since 1 ∈ S , we deduce 1 + 1 = 2 ∈ S . Now apply (ii) with n = 2: since 2 ∈ S , we deduce 2 + 1 = 3 ∈ S . Now apply (ii) with n = 3: since 3 ∈ S , we deduce 3 + 1 = 4 ∈ S . And so forth. This is not a proof. (No good proof uses “and so forth” to gloss over a key point!) But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that every positive 1 2 PETE L. CLARK integer can be reached in this way, i.e., starting from 1 and adding 1 suﬃciently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical induction. Alternately, the principle of mathematical induction is a key ingredient in any ax- iomatic characterization of the natural numbers. It is not a key point, but it is somewhat interesting, so let us be a bit more spe- cific. In Euclidean geometry one studies points, lines, planes and so forth, but one does not start by saying what sort of object the Euclidean plane “really is”. (At least this is how Euclidean geometry has been approached for more than a hundred years. Euclid himself gave such “definitions” as: “A point is that which has posi- tion but not dimensions.” “A line is breadth without depth.” In the 19th century it was recognized that these are descriptions rather than definitions, in the same way that many dictionary definitions are actually descriptions: “cat: A small car- nivorous mammal domesticated since early times as a catcher of rats and mice and as a pet and existing in several distinctive breeds and varieties.” This helps you if you are already familiar with the animal but not the word, but if you have never...
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