Prove that irrational numbers are dense in the number line?

Depends on what starting arguments (or axioms) you are allowed to use.

A typical "proof" based on the Rationals which are dense, 

then the set {Rationals + √2 } is dense. But this set is a subset of the Irrationals, hence the Irrationals is dense. 

Another typical proof is that taken any irrational number R and any real number e>0 construct an irrational number inside the interval [R, R+e]. This is trivial to construct.