I am having trouble proving the statement:

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

**Answer**

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

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