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Problem # 3 These problems concern approximating arbitrary real numbers by certain kinds of sequences. a) Prove that for every real number r 2 R there exists a sequence of rational numbers q1 < q2 < : : : < r with the property qn ! r. b) Prove that for every real number r 2 R there exists a sequence of irrational numbers z1 < z2 < : : : < r with the property zn ! r. Problem #4 Let an > 0 be a sequence of strictly positive numbers. Suppose that = lim an+1 an exists. a) Prove that if < 1 then an ! 0. b) Prove that if > 1 then an ! 1. c) Give an example with = 1 and an ! 0. d) Give an example with = 1 and an ! 1. e) Give an example with = 1 and an bounded but not converging to any limit at all. Problem #5 Let a1; a2; : : : aN be a collection of real numbers. A weighted average is an expression of the form: k1a1 + k2a2 + : : : kNaN AN = ; k1 + k2 + : : : + kN where all ki 2 N. Note that the case k1 = k2 = : : : = kN = 1 corresponds to the usual notion of average. a) Prove that if an ! L then the weighted averages above are such that AN ! L regardless of how the weights ki are chosen. b) Show that a similar result is true when an ! +1.

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