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Math Help - Proof involving rational numbers

  1. #1
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    Proof involving rational numbers

    a) Prove that between any two rational numbers there is another rational number; that is, if a, b are in Q and a < b, then there exists z in Q such that a < z < b.

    b) Prove that between any two rational numbers there are infinitely many rational numbers.
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  2. #2
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    Proof that there is a rational between any two real numbers

    Since a<b then

    0< \frac{1}{b-a} by the archimedian principle we can pick an Integer N such that

    0< \frac{1}{b-a}<N This implies that

    1 < N(b-a) we can now pick another integer n such that

    n < N(a) < n+1 adding 1 we get

    n+1 < N(a)+1 usin g the fact that 1 < N(b-a) we get

    n+1 <N(a) +1 < N(a) +N(b-a)=N(b) this implies that

    N(a) < n+1 < N(b) \implies a < \frac{n+1}{N}< b
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  3. #3
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    You don't have to be that complicated! If a and b are rational numbers, then c= \frac{a+ b}{2} is a rational number between a and b.

    Suppose there were only a finite number of rational numbers between a and b. Then there must be largest such number: c. But (b+c)/2 is a rational number between c and b and so between a and b and is larger than c: contradiction.
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