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 Rtheorem...
Without loss of generality suppose that the interval is .
Define by .
Then, , which implies has infinitely many rational numbers.
Here, I used the convension .
Extension:
Let be any two real numbers, then we may find (because of ) such that , and set for .