Task: Give an example of metric space in which not every Cauchy sequence converges.
My Work:
Consider the metric space $X=\mathbb{Q}$ with the metric $d(x,y)=|x-y|$.
Now $a_n = \left(1+\frac{1}{n}\right)^n$ is Cauchy sequence and $\lim a_n=e\notin \mathbb{Q}$.
Does this work to show that not every Cauchy sequence in $\mathbb{Q}$ converges?
Your example is indeed correct. You could elaborate more on why it is Cauchy and has no limit.
Probably the simplest argument is that it is well known that the sequence converges to $e$ in $\mathbb R$. As a converging real sequence it is Cauchy. If it had a limit, it would be the same as the limit on the real line, as the metrics are the same. But $e\notin\mathbb Q$, so there is no limit.