By Asuman G. Aksoy

Schooling is an admirable factor, however it is definitely to recollect at times that not anything worthy understanding might be taught. Oscar Wilde, “The Critic as Artist,” 1890. research is a profound topic; it truly is neither effortless to appreciate nor summarize. in spite of the fact that, actual research will be found through fixing difficulties. This ebook goals to offer self sustaining scholars the chance to find actual research by means of themselves via challenge fixing. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental heritage. even supposing research used to be conceived within the seventeenth century in the course of the Scienti?c Revolution, it has taken approximately 200 years to set up its theoretical foundation. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz have been between those that contributed to its genesis. Deep conceptual alterations in research have been led to within the nineteenth century through Cauchy and Weierstrass. moreover, sleek innovations similar to open and closed units have been brought within the 1900s. this day approximately each undergraduate arithmetic application calls for at the least one semester of genuine research. usually, scholars ponder this path to be the main difficult or perhaps intimidating of all their arithmetic significant requisites. the first target of this ebook is to relieve these issues via systematically fixing the issues relating to the center options of such a lot research classes. In doing so, we are hoping that studying research turns into much less taxing and thereby extra pleasant.

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A. 1007/978-1-4419-1296-1_2, © Springer Science+Business Media, LLC 2010 CHAPTER 2. REAL NUMBERS 22 strong induction it is not necessary to list the basis step, it is clearly true that the statement holds for all previous cases. The inductive step of a strong induction in this case corresponds to the basis step in ordinary induction. • Let A be a nonempty subset of R. The number b is called an upper bound for A if for all x ∈ A, we have x ≤ b. A number b is called a least upper bound of A if, ﬁrst, b is an upper bound for A and, second, b is less than or equal to every upper bound for A.

Easy algebra manipulations give a 1 a 1 < ≤ . m+1 b m If 1 b = , then we must have m > 1. In this case take n = m − 1. Otherwise take n = m to get a m 1 a 1 ≤ < . n+1 b n To prove the second part we will use the strong induction argument. If a = 1, then the conclusion is obvious. Assume the conclusion is true for a = 1, . . , k and let us prove that the conclusion is also true for a = k + 1. Let b ≥ 1 be a natural number coprime with k + 1 such that k + 1 < b. Then, the ﬁrst part implies the existence of a natural number n ≥ 1 such that 1 k+1 1 ≤ < .

Hence the inequality is also true for n + 1. Therefore, by induction we have (1 + x)n ≥ 1 + nx for all natural numbers n ≥ 0. 7 The classical induction argument will not work here. The main reason is that in order to reach Fn+2 one will need to make assumptions about Fn+1 and Fn . Therefore, we will use a strong induction argument. Indeed, ﬁrst it is obvious that √ √ (1 + 5)1 − (1 − 5)1 √ . F1 = F2 = 21 5 Next assume that Fk = (1 + √ 5)k − (1 − √ 2k 5 and let us prove that √ √ 5)k , k = 1, 2, .