Real Analysis/Section 1 Exercises/Answers

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Real Analysis/Section 1 Exercises


  1. Show that -1<0\
  2. Show that \forall x,y\in\mathbb{Z},\ -(xy)=(-x)y=x(-y)
  3. Show that \forall z\in\mathbb{Z},\ z^2\ge0
  4. Show that \forall z\in\mathbb{Z},\ z\ge0\Leftrightarrow-z\le0
  5. Show that \forall x,y,z\in\mathbb{Z},\ (x<0)\cap(y<z)\Leftrightarrow(xy>xz)
  6. Let p be any prime. Show that \sqrt{p} is irrational.
  7. Complete the proofs of the simple results given above.
  8. Show that the complex numbers \mathbb C cannot be made into an ordered field.
  9. Complete the proof of the square roots theorem by giving details for the case x<1.
  10. Suppose A is a non-empty set of real numbers that is bounded above and let s = sup A. Show that if s is not in A, then for any ε > 0, there exists an element a in A such that s − ε < a < s.

Hints / Answers[edit]

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  6. With the view of getting a contradiction, assume \sqrt{p} is rational. Then \sqrt{p}=r/s for some integers s and r such that they have no common factor other than one, (that is, s and r are in lowest terms). Squaring both sides and rearranging terms gives s^2p=r^2. Since p is prime, r must be divisible by p, say r=pt for some integer t. By substitution, s^2p=p^2t^2, so that s^2=pt^2, and thus s must also be divisible by p, a contradiction.
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