Three nonnegative real numbers r1, r2, r3 are written on a blackboard. These numbers have the property that there exist integers a1, a2, a3, not all zero, satisfying

a1 r1 +a2 r2 +a3 r3 = 0.

We are permitted to perform the following operation: find two numbers x, y on the blackboard with x <= 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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Three nonnegative real numbers r1, r2, r3 are written on a blackboard. These numbers have the property that there exist integers a1, a2, a3, not all zero, satisfying

a1 r1 +a2 r2 +a3 r3 = 0.

We are permitted to perform the following operation: find two numbers x, y on the blackboard with x <= 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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