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A proof in real Analysis

  1. Apr 8, 2010 #1
    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 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 =)
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 8, 2010 #2
    If [itex]\epsilon[/itex] is not a rational number, then is [itex]\epsilon/n[/itex] rational?

    If [itex]|x-y|>\epsilon[/itex] then can you find a rational number such that [itex]\epsilon[/itex] is larger than this rational number? The rest of your arguments can be used provided you find this rational number.
    Last edited by a moderator: May 4, 2017
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