If you "would need the fact that is dense in " then there's already nothing to prove.

You can try the following: if is rational there's nothing to prove, otherwise we can write , an infinite non-periodic decimal expression, so you can now define . It's now simple to check that .

The above is somewhat unsatisfying though, since we have first to know the decimal expression of the irrational in order to be able to define the rational sequence that converges to it.

Tonio