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Unformatted text preview: Math 3110 Homework 1 Solutions Exercise 2.1.1 If a n and b n are increasing, then for all natural numbers n , a n +1 a n and b n +1 b n . This means that a n +1 + b n +1 a n + b n +1 a n + b n for all n , so { a n + b n } is also increasing. The sequence { a n b n } does not have to be increasing, however. Consider a n = n and b n = 2 n . This gives a n b n = n , which is not increasing. Exercise 2.4.7 We prove the extended triangle inequality by induction on n . The base case of n = 2 is the ordinary triangle inequality. (If you wish to extend this to the cases of n = 0 or n = 1, these are trivial, but do not work well for induction.) For the inductive hypothesis, we assume  a 1 + a 2 + + a n   a 1  +  a 2  + +  a n  . In the inductive step, we use the triangle inequality and then the inductive hypothesis to get  a 1 + a 2 + + a n + a n +1   a 1 + a 2 + + a n  +  a n +1   a 1  +  a 2  + +  a n  +...
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 '08
 RAMAKRISHNA
 Natural Numbers

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