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Thread: real analysis

  1. #1
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    real analysis

    prove that for each $\displaystyle x \in \mathbb{ R} \mbox { and each} \ n \in \mathbb{ N }$
    $\displaystyle \mbox { there is rational number} \\ $ $\displaystyle \mbox {rn such that } \ \mid r n $ $\displaystyle - x \mid< \frac{1}{n} $
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  2. #2
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    Quote Originally Posted by flower3 View Post
    prove that for each $\displaystyle x \in \mathbb{ R} \mbox { and each} \ n \in \mathbb{ N }$
    $\displaystyle \mbox { there is rational number} \\ $ $\displaystyle \mbox {rn such that } \ \mid r n $ $\displaystyle - x \mid< \frac{1}{n} $
    Is there a rational number between these two numbers $\displaystyle \frac{x}
    {n} - \frac{1}
    {{n^2 }}\;\& \,\frac{x}
    {n} + \frac{1}
    {{n^2 }}?$
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  3. #3
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    Is there a rational number between these two numbers sure!!!
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  4. #4
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    Quote Originally Posted by flower3 View Post
    Is there a rational number between these two numbers sure!!!
    Well call it $\displaystyle r$.
    Then you are done if you multiply by $\displaystyle n$ and write the interval in absolute value form.
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