$(a_n)_{n \geq 1}=\mathbb{Q}_+$ and $\sqrt[n]{a_n}$ is convergent 
Is there any sequence $(a_n)_{n \geq 1}$ such that it contains all positive rational numbers without repetition, and $\sqrt[n]{a_n}$ is convergent?

My first guess is that there is no such sequence. I tried to build $a_n$ just like the sequence in the proof that $\mathbb{Q}$ is countable: $$1,2,1/2,1/3,3,4,3/2,2/3,1/4,1/5,2/4$$ and so on. I'm not sure if this works, however

Actually the standard one 
$$ (a_n) = \left( \frac 11, \frac 21, \frac 12, \frac 31, \frac 13, \frac 41, \frac 32, \frac 23, \frac 14, \cdots\right) $$
works. The observation is that for the members $a_n$ in the $i$-th layer: 
$$ \frac i1, \frac{i-1}{2}, \cdots, \frac{2}{i-1}, \frac 1i,$$
we have 
$$ i \ge a_n \ge i^{-1}\Rightarrow \sqrt[n]{i} \ge \sqrt[n]{a_n} \ge (\sqrt[n]{i})^{-1}.$$
But clearly $n\ge i$, so 
$$ \sqrt[n]{n} \ge \sqrt[n]{a_n} \ge (\sqrt[n]{n})^{-1}.$$
So $\sqrt[n]{a_n} \to 1$ as $\sqrt[n]{n} \to 1$. 
