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### Intuition for a proof that the rationals are incomplete. [duplicate]

Let A be a set of positive rationals $p$ such that $p^2<2$. Now this set contains no upper bound. To prove this, for every rational $p$, a number $p- \frac{p^2-2}{p+2}$ is associated. This number (...
288 views

### Choice of $\xi$ [duplicate]

Possible Duplicate: Rational Numbers Suppose $\{x \in \mathbb{Q}|x>0,x^2<2\}$ has a supremum. Call this supremum $c$. In order to show that this cannot be the case, we learned that we need ...
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### Constructing a larger rational whose square is also less than two [duplicate]

Possible Duplicate: Rational Numbers Baby Rudin has a very nice construction showing that, given a positive rational number whose square is less than (greater than) two, one can always find a ...
I understand the following proof but how did the author come up with the following expression? $$q = p - \frac{p^2 -2}{p + 2}$$ $q$ being defined that way is crucial for the proof but what's so ...