Are Odd Rationals Dense on Intervals?

[cragar]

Homework Statement

Is the Set X Dense on any interval between (0,1)
X= $\{ \frac{p}{q} \}$ where p and q are odd positive integers with
p<q

The Attempt at a Solution

so we know that q is always bigger than p so it will always be less than 1.
and since p and q are odd we will not have the rationals that have even factors.
So I do not think it will be dense anywhere. The rationals are dense in the reals but we only have odd numbers divided by odd numbers.

 

[Dick]

If you pick a rational p/q then I think (2^n*p+1)/(2^n*q+1) is awfully close to p/q for n large.

 

[cragar]

so should I look at the limit of that.

 

[Dick]

so should I look at the limit of that.

Suppose you did, what would that tell you?

 

[cragar]

well it would go to zero because the bottom will grow faster than the top. But it seems like it would have a chance of maybe being dense close to zero.

 

[Dick]

well it would go to zero because the bottom will grow faster than the top. But it seems like it would have a chance of maybe being dense close to zero.

I don't think it goes to zero.

 

[cragar]

ok, but with p<q I could make q as large as I want and keep p small.

 

[Dick]

ok, but with p<q I could make q as large as I want and keep p small.

No! Fix p and q. Show there is a rational number of the form odd/odd that is as close to p/q as you want.

 

[cragar]

if p and q are fixed then that limit should go to 1.

 

[Dick]

if p and q are fixed then that limit should go to 1.

What limit goes to 1? I really don't know what you are talking about.

 

[cragar]

your post # 2 , as n goes to infinity , that whole formula should go to 1.

 

[Dick]

your post # 2 , as n goes to infinity , that whole formula should go to 1.

No, it does not. It goes to p/q. That was my whole point!