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ma017-2010-hw1 - MA017 2010 HOMEWORK 1 Due Tue n(1 Let a1...

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MA017 2010 HOMEWORK 1 Due 2010-10-05 Tue. (1) Let a 1 , . . . , a n be real numbers, and define f ( x ) = n j =1 a j sin( jx ). Suppose | f ( x ) | ≤ | sin x | for all real x . Prove, by induction , that | n j =1 ja j | ≤ 1. (Considering f 0 (0) gives a “slick” proof, but I am asking you to use induction.) (2) Show that for any set of ten positive integers less than one hundred, there are two disjoint nonempty subsets having the same sum. (3) Let f 0 ( x ) = 1 / (1 - x ), and define f n +1 ( x ) = xf 0 n ( x ). Prove that f n ( x ) > 0 for 0 < x < 1 and all n 0. (4) Given a set of 70 distinct positive integers not exceeding 200, show that two of them differ by 4, 5 or 9. (5) Given any positive integer k 2, show that we can find a sequence of integers
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