a3 - Department of Mathematics and Statistics McGill...

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Department of Mathematics and Statistics McGill University MATH 242 Assignment 3 Due in class, Friday October 23 Full credit can be obtained from correct answers to questions totalling 70. 1. (10 points) Let a > 0 and suppose that x 3 1 > a . We define x n +1 = 2 x 3 n + a 3 x 2 n and verify that x n +1 > 0 inductively. (i) Show that x 3 n +1 - a = (8 x 3 n + a )( x 3 n - a ) 2 27 x 6 n and x n +1 - x n = - x 3 n - a 3 x 2 n . (ii) Show that x 3 n > a for all n N and that ( x n ) is a decreasing sequence bounded below by zero. (iii) Deduce that ( x n ) converges to some real number x 0 . Show that x 3 = a . 2. (10 points) Prove or disprove by counterexample the following statement. If I j are open intervals for j N and Q [ j =1 I j , then [ j =1 I j = R . 3. (10 points) Let ( x n ) n =1 be a sequence of real numbers. If x kn + ` converges as n → ∞ for all integers k and ` with k 2 and ` 1 , show that x n converges. 4. (10 points)
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a3 - Department of Mathematics and Statistics McGill...

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