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hw1sol - Math 3110 Homework 1 Solutions Exercise 2.1.1 If...

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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 | + | a n +1 | . Therefore, by induction, the extended triangle inequality holds.
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