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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 eld 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 eld 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, Equations, Bernoulli

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