I want to prove $\mathbb{Q}$ dense in $\mathbb{A}_\mathrm{fin}=\prod_{p\ \mathrm{finite\ places}}'\mathbb{Q}_p$. $\prod_{p\ \mathrm{finite\ places}}'\mathbb{Q}_p$ means for elements $(x_p) \in \mathbb{A}_\mathrm{fin},\ x_p \in \mathbb{Q}_p$ and for all but finitely $p$, $x_p \in \mathbb{Z}_p$.
I can show that $\forall (x_p) \in \mathbb{A}_\mathrm{fin},\ \forall \epsilon_p>0$ and $\epsilon_p=1$ for all but finitely $p$, $\exists\ x \in \mathbb{Q}\ s.t.\ |x-x_p|<\epsilon_p$. The difficulty is how to prove without the condition "$\epsilon_p=1$ for all but finitely $p$". Since the primes are infinite, the chinese reminder theorem doesn't work and I got stuck.
Furthermore, it seems that you can remove any place instead of the infinite place in $\mathbb{A}_\mathbb{Q}=\prod_{p\ places}'\mathbb{Q}_p$ and the result still holds. How to prove that?
Thanks for any help!