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May 23rd, 2017, 12:39 AM  #11 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  How about rounding up the smaller number then truncating AFTER the first decimal position where they differ? Your counter example would be done thus: $0.2989989998...$ would be rounded to $0.3000000000... \ \ \ $ < rounded up value of smaller number $0.3010010001... \ \ \ $ < unchanged value of larger number They now differ at the 4th decimal position so $ \frac{0.300 + 0.301}{2} = 0.3005 $ and this is in between the original 2 numbers Looking forward to your response 
May 23rd, 2017, 03:48 AM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra 
My response: Proof That a Rational lies between two Irrationals 

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