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ma8_wk2_wednesday_notes_2010 - MATH 8 SECTION 1 WEEK 2...

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MATH 8, SECTION 1, WEEK 2 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Wednesday, Oct. 6th’s lecture, where we showed that there are infinite sets with different cardinalities. 1. Random Question Question 1.1. Define α ( n ) as the function that takes in an integer and returns the total number of prime divisors of n : for example α (9) = 2 , α (36) = 4 , and α (7) = 1 . Find lim n →∞ α ( n ) n . 2. Introduction The goal of this lecture is to prove the following: Theorem 2.1. The sets N and R have different cardinalities. In our last lecture, we introduced the idea of cardinality, which allows us to make sense of the idea of having “different” sizes of infinity; the last tool we need here before we begin our proof, then, is to define the sets we are working with. The natural numbers are (hopefully, by this point) something we understand quite well: the reals, however, we have not yet given a rigorous footing. Let’s change that. 3. Defining the Real Numbers In Q , there are sequences { a n } n =1 of rational numbers that have the three fol- lowing properties: the terms a n are increasing, there is an upper bound for the terms a n , but this sequence of a n ’s doesn’t converge to a rational number. Specifically, consider the following example: a 0 = 1 a 1 = 1 . 4 a 2 = 1 . 41 a n = the first n decimal places of 2; i.e. a n = b 2 · 10 n c 10 n Proposition 3.1. The sequence { a n } n =1 defined above converges to 2 , an irra- tional number. 1
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2 TA: PADRAIC BARTLETT Proof. Before we begin our proof, we should probably say just what convergence *is:* Definition 3.2. A sequence { a n } n =1 converges to some value λ if, for any distance , the a n ’s are eventually within of λ . To put it more formally, lim n →∞ a n = λ iff for any distance , there is some cutoff point N such that for any n greater than this cutoff point, a n
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