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Proof that Q is dense in R

  1. Dec 23, 2012 #1
    I have a difficulty in understanding the proof on Rudin's book page 9 regarding the density of Q in R. Specifically, I don't understand this step:

    After we prove that there exist two integers [itex]m_{1}[/itex], [itex]m_{2}[/itex] with [itex]m_{1}>nx[/itex] and [itex]m_{2}>-nx[/itex] such that:

    [itex]-m_{2}<nx<m_{1}[/itex]​

    What I don't understand is how from the above get's concludes the following:

    Hence there is an integer m (with [itex]-m_{2}≤m≤m_{1}[/itex]) such that:

    [itex]m-1≤nx<m[/itex]​
     
  2. jcsd
  3. Dec 23, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    If m1-1 <= nx, then set m=m1 and both inequalities are trivial.
    If m1-1 > nx, consider m1-2 and so on.
    As the difference between m2 and m1 is finite, you find m in a finite number of steps.
     
  4. Dec 23, 2012 #3

    Bacle2

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    Science Advisor

    Another approach:

    Show every Real number is the limit of a sequence of rationals:
    For rationals, use the constant sequence; for irrationals x, use the decimal
    approximation of x, and cut it at the n-th spot (and apend 0's to the right), i.e.

    x=ao.a1a2....am... --> x':=ao.a1a2.....am00000....0....

    Then x' is rational, and |x-x'|< 10^{-m}

    For any accuracy you want, adjust m, i.e., let it go as far as you want.
     
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