I need help proving that between any two real numbers there exists a rational number.

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- October 22nd 2009, 05:18 PMspikedpunchRational number
I need help proving that between any two real numbers there exists a rational number.

- October 22nd 2009, 09:09 PMtonio
- October 23rd 2009, 07:36 AMHallsofIvy
Tonio's proof is that between any two

**rational**numbers, there exist an irrational number. If x is NOT rational, It's a little more complicated.

Suppose that r and s are the given real numbers, with r< s. Again, look at t= (r+s)/2. That is a**real**number between r and s but may not be rational. Let . There exist an increasing sequence of rational number converging to any real number (The sequence got by truncating the decimal expansion of the number at the decimal place is such a sequence.) so there exist an increasing sequence of rational numbers, converging to t. Since it converges to t, there exist N such that if n> N . for n> N is then larger than t- (s- r)/2= (s+r)/2- (s- r)/2= r and is less than t< s because it is an increasing sequence. - October 23rd 2009, 10:07 AMtonio
- October 24th 2009, 12:12 AMproscientia
All

__tonio__has shown is that between two rational numbers there exists a rational number; this does not imply that between two**real**numbers there exists a rational number.

More precisely, we want to show that between two*distinct*real numbers there exists a rational number. Let with Then with by the Archimedean principle. Let (so is the largest integer such that Then and also (otherwise Hence

and