real analysis

**flower3** · August 21st 2009, 09:12 AM

prove that for each $x \in \mathbb{ R} \mbox { and each} \ n \in \mathbb{ N }$
$\mbox { there is rational number} \\$ $\mbox {rn such that } \ \mid r n$ $- x \mid< \frac{1}{n}$

**Plato** · August 21st 2009, 09:25 AM

Originally Posted by flower3

prove that for each $x \in \mathbb{ R} \mbox { and each} \ n \in \mathbb{ N }$
$\mbox { there is rational number} \\$ $\mbox {rn such that } \ \mid r n$ $- x \mid< \frac{1}{n}$

Is there a rational number between these two numbers $\frac{x}<br /> {n} - \frac{1}<br /> {{n^2 }}\;\& \,\frac{x}<br /> {n} + \frac{1}<br /> {{n^2 }}?$

**flower3** · August 21st 2009, 12:36 PM

Is there a rational number between these two numbers

sure!!!

**Plato** · August 21st 2009, 01:26 PM

Originally Posted by flower3

Is there a rational number between these two numbers

sure!!!

Well call it $r$ .
Then you are done if you multiply by $n$ and write the interval in absolute value form.

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