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 two reals strictly positive such as 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 a natural number. With the help of the last question, demonstrate that for all r>0, it exists two relatives number p and q such as:
p>=N and |ap+bq|<= r
We can use the set F={ap+pq, such a |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
I'm here to ask you your help.
I have to do an exercise but I have difficulties to finish.
Let a and b two reals strictly positive such as 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 a natural number. With the help of the last question, demonstrate that for all r>0, it exists two relatives number p and q such as:
p>=N and |ap+bq|<= r
We can use the set F={ap+pq, such a |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