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

I am having trouble proving the statement:

Let $$S={m+n√2:m,n∈Z}S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $$ϵ>0\epsilon > 0$$, the intersection of $$SS$$ and $$(0,ϵ)(0, \epsilon)$$ is nonempty.

Hint: $|\sqrt2 -1|<1/2$, so as $n\to\infty$ we have that $(\sqrt2-1)^n\to ?$ In addition to that use the fact that the set $S$ is a ring, i.e. closed under multiplication and addition.