"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?