GCD and Cauchy sequences of rationals

Let be the set of all prime numbers. For any with and , I will use the standard GCD . Let be an arbitrary Cauchy sequence. For notation, . We will call the sequence GCD-Cauchy if it is Cauchy and (all limits must exist, obviously). Given an irrational number , does there exist a GCD-Cauchy sequence of rationals whose limit is ?

Re: GCD and Cauchy sequences of rationals

I am thinking the answer is yes, and if I can show that for any pair of distinct primes , the set is dense in the reals, then let . Now, we define and for each , choose such that . Then, let . From there, I would know how to show that this is a GCD-Cauchy sequence whose limit is . But, this construction relies on that density property I mentioned. I have been trying to think of an argument for it (or against it), but so far I have yet to come up with anything substantial.