# real analysis

• Aug 21st 2009, 09:12 AM
flower3
real analysis
prove that for each $\displaystyle x \in \mathbb{ R} \mbox { and each} \ n \in \mathbb{ N }$
$\displaystyle \mbox { there is rational number} \\$ $\displaystyle \mbox {rn such that } \ \mid r n$ $\displaystyle - x \mid< \frac{1}{n}$
• Aug 21st 2009, 09:25 AM
Plato
Quote:

Originally Posted by flower3
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 $\displaystyle \frac{x} {n} - \frac{1} {{n^2 }}\;\& \,\frac{x} {n} + \frac{1} {{n^2 }}?$
• Aug 21st 2009, 12:36 PM
flower3
Is there a rational number between these two numbers http://www.mathhelpforum.com/math-he...1e076689-1.gif sure!!!
• Aug 21st 2009, 01:26 PM
Plato
Quote:

Originally Posted by flower3
Is there a rational number between these two numbers http://www.mathhelpforum.com/math-he...1e076689-1.gif sure!!!

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