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# hw1 - x< ± for every ±> 0 then x ≤ 0(8 Complete the...

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Math 230: Homework 1 Due Friday, Sept. 15 (1) Solve problems 3, 5, and 20 from Rudin chapter 1 (pp. 22-23 of the handout). (2) For any a < b in an ordered field, show that a < a + b 2 < b. (3) In an ordered field, given a > 0, prove that x 2 < a 2 is equivalent to - a < x < a . (4) Prove (in Q ) that ( x - 2) 2 < 9 is equivalent to - 1 < x < 5. (5) Prove the arithmetic-geometric inequality , namely that for a > 0 , b > 0 in an ordered field, we have a + b 2 2 ab. (Hint: Use the fact that ( a - b ) 2 0.) (6) Let A = { a Q : a > 0 , a 2 < 2 } and B = { a Q : a > 0 , a 2 > 2 } . (a) If x B , prove that 2 /x A . (b) Prove that the average ( x + 2 /x ) / 2 is again in B . (c) This gives a nice iteration scheme for approximating 2: Start- ing with a 1 = 2 B , we define a 2 = 2 /a 1 = 1, a 3 = ( a 1 + a 2 ) / 2, and so on: a 2 n = 2 /a 2 n - 1 , a 2 n +1 = ( a 2 n + a 2 n - 1 ) / 2 . Prove that 0 < a 2 n +1 - a 2 n +2 1 / 2 n . Compute the sequence by hand through a 8 . (7) Prove, in any ordered field, that if
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Unformatted text preview: x < ± for every ± > 0, then x ≤ 0. (8) Complete the partial discussion in Rudin’s Step 6 (pp. 19-20) estab-lishing multiplication and its properties in R + . (9) Let E and F be nonempty subsets of R . Suppose that for all x ∈ E and y ∈ F we have x < y . Prove that sup E ≤ inf F. Can “ ≤ ” be replaced by “ < ”? (10) Is the set α = { x ∈ Q : x 2 < 3 } a Dedekind cut? If so, prove that α 2 = 3. If not, ﬁnd a Dedekind cut whose square is 3. (11) Theorem 1.20 in Rudin is proven “abstractly” as a consequence of the least-upper-bound property. Give a “concrete” proof using the construction of R by Dedekind cuts. 1...
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