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February 25th, 2018, 01:45 PM   #1
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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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existence, m or 3n∈xr, m∈ℤ, metric spaces, n∈ℕ, open and closed sets, real analysis, x∈ℝ



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