"Halfway" through a proof, help needed

• Oct 30th 2009, 06:04 PM
paupsers
"Halfway" through a proof, help needed
I'm trying to prove that any nonempty open interval (a, b) contains a rational point and an irrational point.

I've been trying to do this by cases, so what I have so far is (I have proven these):
i) if a and b are both rational, there exists an irrational between them
ii) if a and b are both rational, there exists a rational between them
iii) if a and b are both irrational, there exists a rational between them

So, what I have left to prove is:
I) if a and b are both irrational, then there exists an irrational between them
II) if a is rational and b is irrational, then there exists a rational between them
III) if a is rational and b is irrational, then there exists an irrational between them

I know it sounds kinda wordy, but you get the idea, right? My question is whether or not the information I already have (i, ii, and iii) are enough to prove the theorem already... or is there an entirely easier way to go about this?
• Oct 30th 2009, 07:49 PM
proscientia
There is no need to worry about whether $a$ and b are rational or irrational. First show that there is exists a rational number between $a$ and $b.$ See this post.

Having found a rational number $r\in(a,\,b),$ let $t_n=r-\frac{\sqrt2}n.$ Then $\left(t_n\right)_{n\,=\,1}^\infty$ is an increasing sequence of irrational numbers whose limit is $r;$ hence $\exists\,N$ such that $a i.e. $t_N$ is an irrational number between $a$ and $b.$