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December 30th, 2016, 01:57 PM   #1
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dense set

Hello,
I'm here to ask you your help.
I have to do an exercise, but I have difficulties to finish.
Let a and b be two reals strictly positive such that a/b is irrational.
First, I have demonstrated that G=aZ+bZ is dense and then I have demonstrated that if A is a dense set and F a finish set, then A\F is also a dense set.

This is now that I freeze.
Let N be a natural number. With the help of the last question, demonstrate that for all r>0, there exist two relatives number p and q such that:
p>=N and |ap+bq|<= r
We can use the set F={ap+pq, such that |p|<N and |q|<M} for a good choice of M.

Deduce that the set {ap+bq, and p>=N} is dense in R.

My research: I found M=(r+aN)/b
And then I found that the F set is a finish set. But I don't know how to go further.

Thanks for your help.

Last edited by skipjack; December 30th, 2016 at 08:07 PM.
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December 30th, 2016, 08:19 PM   #2
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Should "finish" be "finite"?
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