Hello,

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 ).

Let .

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