Proofs involving real numbers

• Read carefully pages 105 through 108 (section 4.3)
• Some key questions to answer (try these without looking at the book, but after you’ve read the book):
  1. True or False: \(ab>0\) exactly when \(a\), \(b\) are both negative or both positive.
  2. Prove that the product \(xy\) is equal to \(0\) if and only if \(x=0\) or \(y=0\).
  3. Prove that for all real numbers \(x\), \(y\) we have \(\frac{1}{3}x^2 + \frac{3}{4} y^2 \geq xy\).
  4. State and prove the triangle inequality.
  5. Prove that if for a real number \(x\) and for \(r>0\) we have \(|x-1|<1\) and \(|x-1|<r/4\), then it must be that \(|x^2+x-2|<r\).
  6. True or False: If \(|x-1|<1\) then:
     • \(|x - 1| < 2\)
     • \(|x - 1| < 0.5\)
     • \(|x - 0.5| < 1\)
     • \(|x - 0.5| < 0.5\)
     • \(|x| < 1\)
• Practice problems from section 4.3 (page 115): 4.25, 4.29, 4.31, 4.35, 4.39