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Math Help - Rational Numbers

  1. #1
    Senior Member tukeywilliams's Avatar
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    Rational Numbers

    Let  A be the set of all positive rationals  p such that  p^2 < 2 and let  B consist of all positive rationals  p such that  p^2 > 2 . We want to show that there is no maximum element in  A and no minimum element in  B . So:

     q = p - \frac{p^{2}-2}{p+2} = \frac{2p+2}{p+2} . Then  q^{2} -2 = \frac{2(p^{2}-2)}{(p+2)^{2}} .

    First of all, how do we even know to do this? Just by intuition? Why did we set  q = p - \frac{p^{2}-2}{p+2} = \frac{2p+2}{p+2} ? Could we have set it to something else? So we basically have shown that  p < q for every  p in  A , and  q < p for every  p in  B . By the way, this is from Rudin's "Principles of Mathematical Analysis."
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  2. #2
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    Suppose that a \in A then let b = \frac{{4a}}{{a^2  + 2}}<br />
. Now show that b \in A\quad \& \quad a < b.
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