Assume $\sum s_n$ be a convergent series and $s_n$ are non negative for all $n$. Is it true that $\liminf ns_n=0$?
Attempts: Intuitively I guess the answer is yes because if the series is convergent, i think the sequence $s_n$ has to be converge fast enough to 0 so the tail should have no big difference but i don't quite get how to prove it.
Suppose $\liminf_n n s_n = \delta >0$. Since $\liminf_n a_n = \lim_n \inf_{k \ge n} a_k$, we see that there exists some $N$ such that if $n \ge N$, then $n s_n \ge \frac{\delta}{2}$. This implies that $s_n \ge \frac{\delta}{2n}$, and hence $\sum_{n \ge N} s_n\ge \frac{\delta}{2} \sum_{n \ge N}\frac{1}{n}$. However, the right hand side is unbounded, while the left hand side is bounded, hence we have a contradiction. Hence $\liminf_n n s_n = 0$.