Real Analysis!!!

**bearej50** · February 25th 2009, 04:17 PM

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...

**bkarpuz** · February 26th 2009, 12:33 AM

Originally Posted by bearej50

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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