Proving that m+n√2m+n\sqrt{2} is dense in R\mathbb R

I am having trouble proving the statement:

Let S={m+n2:m,nZ} Prove that for every ϵ>0, the intersection of S and (0,ϵ) is nonempty.


Hint: |21|<1/2, so as n we have that (21)n? In addition to that use the fact that the set S is a ring, i.e. closed under multiplication and addition.

Source : Link , Question Author : user11135 , Answer Author : Jyrki Lahtonen

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