Let $\displaystyle P$ be the set of all prime numbers. For any $\displaystyle a,b\in \mathbb{Q}$ with $\displaystyle a = \prod_{p\in P}p^{a_p}, a_p \in \mathbb{Z}$ and $\displaystyle b=\prod_{p\in P}p^{b_p}, b_p \in \mathbb{Z}$, I will use the standard GCD $\displaystyle (a,b) = \prod_{p\in P}p^{\min(a_p,b_p)}$. Let $\displaystyle a: \mathbb{N} \to \mathbb{Q}$ be an arbitrary Cauchy sequence. For notation, $\displaystyle a(n) = \prod_{p\in P}p^{a_p(n)}$. We will call the sequence $\displaystyle a$ GCD-Cauchy if it is Cauchy and $\displaystyle \lim_{n \to \infty} a(n) = \lim_{m\to \infty} \left( \lim_{n\to\infty} {\left(a(n),a(m) \right)} \right)$ (all limits must exist, obviously). Given an irrational number $\displaystyle x$, does there exist a GCD-Cauchy sequence of rationals whose limit is $\displaystyle x$?