Constructive proof that $\liminf na_n = 0$ if the series $\sum_{n=1}^{\infty} a_n$ converges

Question

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.

Answer 1

Note that $$\inf_{n+1 \leq p \leq 2n}\,pa_p \leq 2n\inf_{n+1 \leq p \leq 2n}\,a_p \leq 2\sum_{p=n+1}^{2n}{a_p}.$$

As a consequence, $$\inf_{k \geq n+1}\,ka_k \leq 2\sum_{k > n}{a_k},$$ which proves the conclusion.