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74-hw10 - MATH 74 HOMEWORK 10 DUE MONDAY APRIL 30 In...

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MATH 74 HOMEWORK 10 - DUE MONDAY, APRIL 30 In problems 1-4 we continue the notation from last week: we have a function L : (0 , ) R satisfying (1) L (10) = 1, (2) if 0 < x < y then L ( x ) < L ( y ), and (3) L ( xy ) = L ( x ) + L ( y ) for all x, y (0 , ). Feel free to use things proved on last week’s homework (the problems and solutions are online). For n N , define a n to be the unique number in N 0 satsifying 10 a n 30 n < 10 a n +1 . For example since 10 1 30 1 < 10 2 , we have a 1 = 1, and since 10 2 30 2 < 10 3 , we have a 2 = 2. 1. What are a 3 and a 4 ? There is a relationship between a n and the number of decimal digits of 30 n . What is it? 2. Fix n N . Prove that a n n L (30) and that L (30) < a n +1 n . 3. Define for n N the sequences b n = a n n and c n = a n +1 n . (a) Is the sequence b n bounded above? (b) Is the sequence b n increasing? [Note that a n is.] (c) Is the sequence c n bounded below? (d) Is the sequence c n decreasing? 4. Prove that the sequences b n and c n are both convergent to L (30). [You may use
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