As stated in this question If $\sum a_n$ converges then $\liminf na_n=0$, the proof in the title can be proven if it can be shown that there is a subsequence of $na_n$ that converges to 0. I was curious if anyone can provide a witness for this statement. The answer given in the linked question is a contradiction argument.
Edit: The sequence $a_n$ is positive. I already know that this is true and understand the non-constructive proof for it, but I was wondering if anyone could provide a proof that involves actually constructing a subsequence of $na_n$ that converges to 0.