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hw6 - IE 500 Fall 2009 E Alper Yldrm HOMEWORK 6(due...

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IE 500 – Fall 2009 E. Alper Yıldırım HOMEWORK 6 (due Wednesday, December 23 in class) 1. Let ( x n ) and ( y n ) be two sequences of real numbers. (a) Suppose that i =1 y i is convergent. Suppose also that | x n | ≤ y n for each n N . Prove that i =1 x i is a convergent series. (b) Suppose that y n x n 0 for each n N . Suppose that i =1 x i is NOT a convergent series. Prove that i =1 y i is not a convergent series. 2. (a) Let ( x n ) be a sequence of nonnegative real numbers such that x 1 x 2 . . . 0. Consider the sequence given by y n = 2 n - 1 x 2 n - 1 , n N (i.e., y 1 = x 1 , y 2 = 2 x 2 , y 3 = 4 x 4 , y 4 = 8 x 8 , etc.). Suppose that i =1 y i is convergent. Prove that i =1 x i is convergent. (b) Suppose that x n = 1 /n q for n N , where q R . Prove, using the previous part, that the series i =1 x i is convergent for q > 1. 3. (a) Let ( X, d ) be a metric space. Let D : X × X [0 , ) be defined as D ( x, y ) = 0 if x = y, max { 1 , d ( x, y ) } otherwise.
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