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I'm studying a proof on this book http://www.math.ucsd.edu/~atparris/papers/book.pdf I can't copy here the proof I'm studying because it is a little bit long. I only need help on a little statement. I don't understand in the page 180 of the book why we can choose $\alpha,\alpha^\prime\in\{\alpha_1,\alpha_2,\alpha_3\}$ distinct such that $x_\alpha-x_{\alpha^\prime}\not\in\{x_\beta-x_{\beta^\prime},x_{\beta^\prime}-x_\beta\}$.

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    The set $\{x_{\beta} - x_{\beta'}, x_{\beta'} - x_{\beta}\}$ contains at most one positive number. Since there are at least two positive values of $x_{\alpha_{i}} - x_{\alpha_j}$ for $i,j \in \{ 1, 2, 3\}$, the claim follows.2011-06-08
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    "Since there are at least two positive values..." means that there are $(i,j)$ and $(\bar{i},\bar{j})$ such that $x_{\alpha_i}-x_{\alpha_j}$ and $x_{\alpha_{\bar{i}}}-x_{\alpha_{\bar{j}}}$ are positive. Why does this imply the claim?2011-06-08
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    If I misunderstand your question, please let me know (I have not indeed read the previous 179 pages!). The set $\{z , -z\}$ contains at most one positive value. We have two positive values, say $x$ and $y$. Therefore, at least one of $x$ or $y$ are NOT in the set $\{z, -z\}$.2011-06-08
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    you're right, sometimes my brain doesn't work. How can I prove that there are at least two $x_\alpha-x_{\alpha^\prime}$ distinct?2011-06-08
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    @Jacob; "Help on a proof" is a very unhelpful title. Worse, you've now edited [another question](http://math.stackexchange.com/questions/45345/help-on-a-proof) into having this exact same title. Please make titles informative, don't edit them to make them less informative2011-06-26
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    @Arturo: Then next time you will edit a question of mine, please let me know what you edited.2011-06-27
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    @Jacob: If you click on the "edited xxx ago", you can see the changes made in each edit. For instance, if you click on [the corresponding link](http://math.stackexchange.com/posts/45345/revisions) for the other question, you will see the only change I made was to the title.2011-06-27

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If you accept that there are at least two positive (distinct) values for $x_\alpha - x_{\alpha'}$ then the claim follows since $\{x_\beta - x_{\beta'}, x_{\beta'} - x_\beta\}$ contains at most one positive value. Indeed, take one of the positive values for $x_\alpha - x_{\alpha'}$. If it is not in $\{x_\beta - x_{\beta'}, x_{\beta'} - x_\beta\}$ then chose it, otherwise chose the other one.

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    how can I prove that there are at least two positive (distinct) values for $x_\alpha-x_{\alpha^\prime}$?2011-06-08
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    wlog assume that $x_{\alpha_1} < x_{\alpha_2} < x_{\alpha_3}$. Then $x_{\alpha_3} - x_{\alpha_1}$ and $x_{\alpha_3} - x_{\alpha_2}$ are two distinct values.2011-06-08