"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?

There is no need to worry about whether and b are rational or irrational. First show that there is exists a rational number between and See this post.

Having found a rational number let Then is an increasing sequence of irrational numbers whose limit is hence such that i.e. is an irrational number between and