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Math Help - [SOLVED] Proof dealing with density of rationals in the reals

  1. #1
    Member ilikedmath's Avatar
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    Post [SOLVED] Proof dealing with density of rationals in the reals

    Statement to prove:
    Let x, y be in the reals with x < y.
    If u is a real number with u > 0, show that there exists a rational number r such that x < ru < y.

    We proved something similar to this in class. We proved the density of the rationals in the reals:
    Claim: If x, y are real numbers and x < y, then there exists a rational number such that x < r < y.
    Proof:
    If x < 0 and y > 0 then x < 0 < y, and clearly 0 is rational.
    If x ≥ 0: Since x < y, then 0 < y - x. By Archimedian Property (AP), there exists a natural number such that n(y - x) > 1 (useing "(y - x)" for "x" in AP and "1" for "y" in AP).
    Note n(y - x) > 1 implies that ny > 1 + nx.
    Let A = {k be a natural number: k > nx}. By AP, A is nonempty. By the Well Ordering Principle, A has a smallest element, say m in A is the smallest. Thus, m > nx ≥ m - 1 (m > nx since m is in A; Since m is the smallest, either (m - 1) is in N\A or (m - 1) = 0 if m - 1 and recall nx ≥ 0.
    Note nx ≥ m – 1 implies 1 + nx ≥ m, but also m > nx so we have 1 + nx ≥ m > nx.
    Since ny > 1 + nx, we have ny > m > nx which implies y > m/n > x and m/n is rational.
    If x < 0 and y ≤ 0, apply the above to -x and -y.
    Then yield r is rational with -y < r < -x which implies y > -r > x and -r is rational.
    QED.

    ----
    So can I just basically state that u is a real number and u > 0 and work that into the proof?
    Last edited by ilikedmath; October 3rd 2008 at 03:51 PM. Reason: Newbie
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  2. #2
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    Quote Originally Posted by ilikedmath View Post
    I also posted my question on Cramster since it is easy for plugging in mathematical symbols:
    Why don't you want to learn to use LaTeX?
    Serious posters know how to post questions on the WEB.
    You ought to learn to do it too.
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  3. #3
    Member ilikedmath's Avatar
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    Latex?

    Sorry, I didn't know that. I edited my post. I hope that is better than what I had before.

    Latex? Can I download that from somewhere? I've never heard of it. My bad.
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  4. #4
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    Your question is a direct consequence of Q being dense in R. Use the field properties of R to reach the result.

    Maybe try this one out: Let a,b \in \mathbb{R} such that b > a > 0. Show that there is an irrational c, such that a < c < b.
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