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Let $a$ and $b$ be reals with $a. Show that there are infinitely many rationals $x$ such that $a.

My plan of action was to assume that $x$ is the smallest such rational and find another rational in the interval $(a, x)$, but I am struggling to make it work. A hint will be much preferred to a full solution.

  • 0
    Just take the average of $a$ and $x$. It's not hard to show that's greater than $a$ and smaller than $x$.2012-09-20
  • 4
    @crf The average of $a$ and $x$ need not be rational.2012-09-20
  • 1
    ! sorry. Somehow I missed the whole thing about them being rationals.2012-09-20

6 Answers 6