Let $(x_n)$ be a Cauchy sequence in the metric space $\mathbb{R}$ with the Euclidean metric, with the property that $x_n\in\mathbb{Q}$ for all $n\in\mathbb{N}$.
Is it true that $(x_n)$ in the metric space $\mathbb{Q}\subset\mathbb{R}$ (and it's induced metric) is Cauchy?
Is this trivial or is there something substantial to prove?
The backstory behind this question is to prove a sequence is Cauchy in $\mathbb{Q}$. The sequence of rational numbers converges to an irrational number in $\mathbb{R}$, hence is Cauchy in $\mathbb{R}$, hence is Cauchy in $\mathbb{Q}$?