$(a_n)_{n \geq 1}=\mathbb{Q}_+$ 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

Pick an enumeration of $\mathbb{Q}_{+}$, $(q_i)_{i \geq 1}$.
We will define $a_n$ to be a reordering of $q_i$ such that $\frac{1}{2n}<a_n<2n$. Any such sequence will satisfy $\sqrt[n]{a_n}\rightarrow 1$.
Define $k_n= \min\{k| q_k \in (\frac{1}{2n},2n), k\notin \bigcup_{i=1}^{n-1}I_i   \}$. We can verify that $\bigcup_{i=1}^{\infty}I_i  =\mathbb{N}$, and by construction $I_i\neq I_j$ when $i\neq j$.
Finally, the sequence $a_n=q_{k_n}$ does the job.
Remark: If $\sqrt[n]{a_n}$ converges then the limit must be $1.$
