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MS Exam’n — Analysis — Jan. 15, 2002 Name: 1 Problem 1. Suppose that E is an uncountable subset of R. Let the set M be defined by M = { x ∈ R | (−∞, x] ∩ E is finite or countable} . Prove that if M 6= ∅, then M is bounded above Also show that if b = sup M , then b ∈ M , and for some x, x > b, both sets (−∞, x] ∩ E and [x, +∞) ∩ E are uncountable, still assuming that M 6= ∅. MS Exam’n — Analysis — Jan. 15, 2002 Name: Blank Page 2 3 MS Exam’n — Analysis — Jan. 15, 2002 Name: Problem 2. Suppose that {an } is a decreasing sequence of non-negative ∞ X an converges real numbers: an ≥ an+1 ≥ 0 for n = 1, 2, . . . . Prove that if n=1 then lim nan = 0. n→∞ MS Exam’n — Analysis — Jan. 15, 2002 Name: Blank Page 4 MS Exam’n — Analysis — Jan. 15, 2002 Name: 5 Problem 3. Suppose that f : [0, +∞) →Z R is a continuous function such 1 x f (t) dt = 0. that lim f (x) = 0. Prove that lim x→+∞ x→+∞ x 0 MS Exam’n — Analysis — Jan. 15, 2002 Name: Blank Page 6 MS Exam’n — Analysis — Jan. 15, 2002 Name: 7 Problem 4. Suppose that {fn } is a sequence of continuous functions on the interval [0, 1] which converges pointwise to 0 on [0, 1]; that is for each x ∈ [0, 1] lim fn (x) = 0. n→∞ Show that if, for each x ∈ [0, 1], the sequence {f n (x)} is decreasing, then {fn } converges uniformly to 0 on the interval [0, 1]. MS Exam’n — Analysis — Jan. 15, 2002 Name: Blank Page mscompanal02.dvi 8