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February 25th, 2018, 01:45 PM  #1 
Newbie Joined: Jul 2014 From: Wrexham Posts: 20 Thanks: 0  Existence of an m∈ℤ n∈ℕ such that m/(3^n)∈(xr,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: [x1/(3^n) , x+1/(3^n)]=[((3^n)x 1)/(3^n),((3^n)x +1)/(3^n)]⊂(xr,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? 

Tags 
existence, m or 3n∈xr, m∈ℤ, metric spaces, n∈ℕ, open and closed sets, real analysis, x∈ℝ 
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