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homework2mat25 - 4 Prove that ac = bc and c 6 = 0 imply a =...

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Homework 2 due Thursday August 14th 1. (1.10) Prove (2 n + 1) + (2 n + 3) + (2 n + 5) + · · · + (4 n - 1) = 3 n 2 for all n N . 2. Use induction to prove Bernoulli’s inequality: If 1 + x > 0, then (1 + x ) n 1 + nx for all n N . 3. Polya’s paradox. Find the error in the following proof by induction. Claim: All horses are the same color. Proof: Base case: If there’s only one horse, then all horses are the same color. Inductive step: Suppose that any set of n horses have the same color. Now suppose we have a set of n + 1 horses. Take all but the first horse, i.e. horses { 2 , 3 , . . . , n +1 } . This is a set of n horses, so they each have the same color by the inductive hypothesis. Now, take all but the last horse, i.e. horses { 1 , 2 , . . . , n } . This is also a set of n horses, so by the inductive hypothesis they each have the same color. Because the sets overlap at horses { 2 , 3 , . . . , n } , both sets of horses are all the same color. Therefore, by induction every horse is the same color. Note: clearly NOT all horses in the world are the same color, so there must be an error here!
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Unformatted text preview: 4. Prove that ac = bc and c 6 = 0 imply a = b for a,b,c ∈ Q . Write down each field property (A1-A4, M1-M4, DL) or order property (O2-O6) each time you use it. 5. Prove that || a | - | b || ≤ | a-b | for all a,b ∈ Q . Write down each field property (A1-A4, M1-M4, DL) or order property (O2-O6) each time you use it. 6. Prove: if 0 ≤ x ≤ ± for all ± > 0, then x = 0. 7. (4.14) Let A and B be non-empty subsets of R that are bounded above, and let S = { a + b : a ∈ A,b ∈ B } . Prove that sup S = sup A + sup B . 8. (4.16) Show that for all a ∈ R , sup { r ∈ Q : r < a } = a . 9. In this exercise you will prove the irrational numbers are dense in R . (a) Let x ∈ Q and x 6 = 0. Let y ∈ R \ Q (i.e. y is irrational). Prove that xy is irrational. (b) Let x,y ∈ R so that x < y . Prove that there exists z ∈ R \ Q so that x < w < y . Hints: apply the density of Q in R to x/ √ 2 and y/ √ 2 and then use part (a). 1...
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