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> **Possible Duplicate:**  
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> [Rational Numbers](https://math.stackexchange.com/questions/14970/rational-numbers) 
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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 to introduce $\xi$ with $\xi=\frac{2c+2}{c+2}$ and then find a contradiction. But why this $\xi$? Why not another $\xi$? How do you find this choice?