TITLE: Proving that there exists an irrational number in between any given real numbers
QUESTION [22 upvotes]: Possible Duplicate:
Density of irrationals 

I am trying to prove that there exists an irrational number between any two real numbers a and b. I already know that a rational number between the two of them exists. My idea was to say represent a and b as $a = \sqrt{2} x, b = \sqrt{2} y$ for some $x, y \in \mathbb{R}.$ We know that there exists $\frac{m}{n}$ s.t. $x < \frac{m}{n} < y,$ so multiplying everything by the root of 2, we have $$\sqrt{2}x = a < \frac{\sqrt{2}m}{n} < \sqrt{2}y = b,$$ and so we have an irrational number in between two reals. Is this 'legit?' Are there 'better/'more elegant ways to go about this? I would really appreciate seeing alternatives proofs of this, as many as possible.

REPLY [4 votes]: HINT $\ $ If the non-empty interval I contains only rationals, then repeatedly shifting I left/right by a fixed positive rational less less than the length of I will cover the entire real line with rationals. Hence we conclude that every real is rational, a contradiction.