December 30th, 2016, 12:57 PM  #1 
Newbie Joined: Dec 2016 From: France Posts: 1 Thanks: 0  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 07:07 PM. 
December 30th, 2016, 07:19 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,956 Thanks: 1603 
Should "finish" be "finite"?


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