(adsbygoogle = window.adsbygoogle || []).push({}); 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[itex]|x-y|>\varepsilon[/itex]

let n belongs Z, n>1

obviously,[itex]\varepsilon[/itex] satisfies [itex]x<x+\frac{\varepsilon}{n}<y[/itex]

as there exist infinite numbers for n,

therefore, infinite r satisfy x<r<y

thanks for reading =)

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# Homework Help: A proof in real Analysis

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