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Math Help - Real Analysis!!!

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    Real Analysis!!!

    Prove: If x and y are real numbers with x < y, then there are infinitely many rational numbers in the interval [x, y].
    I know that I have to prove this using mathematical induction. I also think that I need to use the Density of Q in R theorem...
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    Quote Originally Posted by bearej50 View Post
    Prove: If x and y are real numbers with x < y, then there are infinitely many rational numbers in the interval [x, y].
    I know that I have to prove this using mathematical induction. I also think that I need to use the Density of Q in R theorem...
    Without loss of generality suppose that the interval is [0,1].
    Define f:\mathbb{N}\to[0,1] by f(n)=1/n.
    Then, f(\mathbb{N})=\{1,1/2,1/3,\ldots\}\subset[0,1]_{\mathbb{Q}}\subset[0,1], which implies [0,1] has infinitely many rational numbers.
    Here, I used the convension [0,1]_{\mathbb{Q}}:=[0,1]\cap\mathbb{Q}.

    Extension:
    Let x<y be any two real numbers, then we may find z,w\in\mathbb{Q} (because of \overline{\mathbb{Q}}=\mathbb{R}) such that x<z<w<y, and set f(n)=(w-z)/n+z for n\in\mathbb{N}.
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