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Math Help - "Halfway" through a proof, help needed

  1. #1
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    Question "Halfway" through a proof, help needed

    I'm trying to prove that any nonempty open interval (a, b) contains a rational point and an irrational point.

    I've been trying to do this by cases, so what I have so far is (I have proven these):
    i) if a and b are both rational, there exists an irrational between them
    ii) if a and b are both rational, there exists a rational between them
    iii) if a and b are both irrational, there exists a rational between them

    So, what I have left to prove is:
    I) if a and b are both irrational, then there exists an irrational between them
    II) if a is rational and b is irrational, then there exists a rational between them
    III) if a is rational and b is irrational, then there exists an irrational between them

    I know it sounds kinda wordy, but you get the idea, right? My question is whether or not the information I already have (i, ii, and iii) are enough to prove the theorem already... or is there an entirely easier way to go about this?
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  2. #2
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    There is no need to worry about whether a and b are rational or irrational. First show that there is exists a rational number between a and b. See this post.

    Having found a rational number r\in(a,\,b), let t_n=r-\frac{\sqrt2}n. Then \left(t_n\right)_{n\,=\,1}^\infty is an increasing sequence of irrational numbers whose limit is r; hence \exists\,N such that a<t_N<r<b, i.e. t_N is an irrational number between a and b.
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