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Thread: 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 $\displaystyle [0,1]$.
    Define $\displaystyle f:\mathbb{N}\to[0,1]$ by $\displaystyle f(n)=1/n$.
    Then, $\displaystyle f(\mathbb{N})=\{1,1/2,1/3,\ldots\}\subset[0,1]_{\mathbb{Q}}\subset[0,1]$, which implies $\displaystyle [0,1]$ has infinitely many rational numbers.
    Here, I used the convension $\displaystyle [0,1]_{\mathbb{Q}}:=[0,1]\cap\mathbb{Q}$.

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