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 (A1A4, M1M4, DL) or order property (O2O6) each time you use it. 5. Prove that  a    b  ≤  ab  for all a,b ∈ Q . Write down each ﬁeld property (A1A4, M1M4, DL) or order property (O2O6) each time you use it. 6. Prove: if 0 ≤ x ≤ ± for all ± > 0, then x = 0. 7. (4.14) Let A and B be nonempty 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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 Winter '11
 Wiley
 Differential Equations, Logic, Equations, Bernoulli, Inductive Reasoning, Empty set, Natural number, Mathematical logic

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