# Real Analysis!!!

• Feb 25th 2009, 05:17 PM
bearej50
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...
• Feb 26th 2009, 01:33 AM
bkarpuz
Quote:

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