# s10 - Math 5615H Introduction to Analysis I Fall 2011...

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Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #10. Problems and Solutions. #1. The sequence { a n } is defined by a 1 = 0 , and a n +1 = 2 a n / 2 for n = 1 , 2 , 3 ,.... Prove that { a n } is convergent and find its limit. You can use without proof the fact that the function y = f ( x ) = a x , where a > , a 6 = 1, is strictly convex , i.e. f ( tx 1 + (1- t ) x 2 ) < tf ( x 1 ) + (1- t ) f ( x 2 ) for all x 1 < x 2 and 0 < t < 1 . Solution. By induction, a 1 = 0 < a 2 = 1 < a 3 < ··· < a n < ··· < 2 for all n = 1 , 2 , 3 ,... . By Theorem 3.14 and its proof, there exists L = lim a n = sup { a n } ≤ 2. Moreover, a n +1 = 2 a n / 2 = ⇒ a n +1 ≤ 2 L/ 2 , L ≥ 2 a n / 2 for n = 1 , 2 , 3 ,... = ⇒ L ≤ 2 L/ 2 , L ≥ 2 L/ 2 = ⇒ L = 2 L/ 2 ≤ 2 . We claim that L = 2, i.e. the case L = 2 L/ 2 < 2 is impossible. Indeed, if this is the case, then the equation x = 2 x/ 2 has at least 3 distinct solutions x 1 = L < x = 2 < x 2 = 4. In other words, the graph of a strictly convex function...
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s10 - Math 5615H Introduction to Analysis I Fall 2011...

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