If , x < y. Show that if, x and y are rational, then an irrational number u, x < u < y.
I am not sure what to do now.
Contradiction, contrapositive??
Here's a simple solution using basic set theory:
The interval (x,y) is uncountable.
Since there are only countably many rational numbers, there must be an irrational number in the interval.
Note: this proof doesn't require x and y to be rational, only that x<y.
While DrSteve's answer is very elegant and is how I'd answer it if asked, you may not have those concepts available to you yet. Since the sum of a rational and an irrational is irrational, it suffices to find an irrational number between 0 and y-x. Look at the "Archimedean property" of the reals. Is there an n such that ?