Existence of an m∈ℤ n∈ℕ such that m/(3^n)∈(x-r,x+r) x∈ℝ
I have shown that the set D={m/(3^n):m∈ℤ and n∈ℕ} is countable. I have also shown that there is r>0 such that 1/(3^n) <r for n∈ℕ. My thoughts are: [x-1/(3^n) , x+1/(3^n)]=[((3^n)x -1)/(3^n),((3^n)x +1)/(3^n)]⊂(x-r,x+r). So I need to show that between [(3^n)x -1)/(3^n),((3^n)x +1)/(3^n)] there is always an integer m such that m/(3^n) is in this interval. How would I go about showing that such an m exists?
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