The problem in your proof is, for example in case 1 : how can you say that there exists between x and y ?
In fact, this is where the Archimedian property of intervenes
Here is what I have for dense in
We know that there exists an irrational in , let's call it .
Let such that
Since is Archimedian, there exists such that (both and belong to ).
We have :
Since x and q are rationals and is irrational, z is irrational (a proof by contradiction is suitable. And I know I replied to this a while ago on MHF)
Therefore, we can find an irrational between any 2 rationals (which are real numbers).
Hence is dense in