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Unformatted text preview: satisfy the integer definition. PROVE: If X n = 2n − 1
, then limn → ∞ X n = 2 . n Answer. We begin by computing the "error": €
 €n – A =  2 – (1/n) – 2 = 1/n. X Then given an ε > 0, the goal is to make this error < ε , so in particular we need (1/n) < ε . Bu t this is true if 1/ε < n. Then for this ε > 0, let N = 1/ε , then for any n > N, 1/n < 1/N, so  Xn – A = 1/n < 1/N < ε. Thus the definition is satisfied Problem 9
6: Suppose that Zn is a sequence for which this is true: There is a positive integer N such that for every ε > 0, for all n > N,  Zn – A < ε. What would an example of such a Zn be? What can you prove about Zn that must be true. Answer. This definition says that for all n > N that for that particular n,  Zn – A < ε for any ε > 0. But this means that it must be true that  Zn – A ≤ 0, for if the number were positive, there would be a positive ε that would be smaller. But also  Zn – A ≥ 0 since it is an absolute value,...
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 Winter '11
 JamesKing
 Math, Integers

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