Question 442256: If x < y, then x<(x+y)/2 < y. Use this fact to give a convincing argument for the following: Between any two rational numbers there are infinitely many rational numbers.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Keep in mind that if x and y are rational, then  is also a rational number. The proof is quite simple; simply let  and  , where a,b,c,d are integers and b and d are nonzero. Then
 . This is a ratio of two integers, so the average of x and y is a rational number.
Hence, if we let  , we may let  , and in general,  . All of these numbers are rational, and we can go on indefinitely, so there must be infinitely many rational numbers, regardless of how far x and y are.
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