Euclidean algorithm with reals ?

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rah4927
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Euclidean algorithm with reals ?

Unread post by rah4927 » Thu Nov 10, 2016 1:09 pm

$4.$ Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.

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