# A proof in real Analysis

1. Apr 8, 2010

### Shing

1. The problem statement, all variables and given/known data
If x and y are arbitrary real numbers. x>y. prove that there exist at least one rational number r satisfying x<r<y, and hence infinitely.

3. The attempt at a solution
well, I have done my proof, but comparing to the solution offered by http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-014Calculus-with-Theory-IFall2002/1C8FA521-FDCE-491B-8689-955B04A4A4A2/0/pset2solutions.pdf" [Broken] (*1), I have a bit doubt about whether my proof is precise enough or not.

anyway, here it is:

x,y belong to R, x<y
let$|x-y|>\varepsilon$
let n belongs Z, n>1
obviously,$\varepsilon$ satisfies $x<x+\frac{\varepsilon}{n}<y$
as there exist infinite numbers for n,
therefore, infinite r satisfy x<r<y

If $\epsilon$ is not a rational number, then is $\epsilon/n$ rational?
If $|x-y|>\epsilon$ then can you find a rational number such that $\epsilon$ is larger than this rational number? The rest of your arguments can be used provided you find this rational number.