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# MATH 147 - Homework 2 Due in class Monday, September 28 1. Let a1 = 1, and for each n 1 dene an+1 = 3 + 2an So, for example, a2 = 3 + 2a1 = 5. Prove...

Questions 3&5 in this real analysis assignment. this is a first year course

MATH 147 – Homework 2 Due in class Monday, September 28 1. Let a 1 = 1, and for each n 1 deﬁne a n +1 = 3 + 2 a n So, for example, a 2 = 3 + 2 a 1 = 5. Prove that for every integer n 1, we have 0 a n a n +1 3 2. Let a > 0 be a positive real number. Let A = { a n | n N } = { a,a 2 ,a 3 ,... } be the set of all positive integer powers of a . (a) If a > 1, show that A is not bounded above. [Hint: Use the Binomial Theorem on (1 + ( a - 1)) n .] (b) If a < 1, show that inf( A ) = 0. [Hint: Use part (a) on 1 /a .] 3. Let A be the set of all the real numbers a n from problem 1. Prove that sup( A ) = 3. [Hint: Deﬁne α n = 3 - a n , and use the result from problem 2 to show that for every ± > 0, there is a positive integer n such that α n < ± .] 4. For every integer n 1, deﬁne a n = 1+ 1 1! + 1 2! + ... + 1 n ! . (If k is a positive integer, we deﬁne k ! (pronounced “ k factorial”) to be k ! = 1 · 2 ··· ( k - 1) · k .) Let A be the set consisting of all the a n . Prove that A is bounded above by 3. (You may assume – without proof – standard facts about geometric series. Like geometric series with common ratio 1 / 2, for example.) [The fact that A is bounded above means that A has a supremum. This supremum is called e , and is the base of the natural logarithm. It’s also not bigger than 3.] 5. Let A be the set of 10-adic numbers. That is, let A be the set of inﬁnite decimal expansions ...a n a n - 1 ...a 0 .a - 1 a - 2 ...a - m where the a i are integers between 0 and 9, inclusive, and the expansions are allowed to extend inﬁnitely to the left, but have only ﬁnitely many digits after the decimal point. Two 10-adic numbers are added and multiplied in the usual way, from right to

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