If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum?
I tried to prove as follows: Since $\liminf\, |a_n|=0,$ then we can find $n_1<n_2<n_3\ldots$ such that $|a_{n_{1}}|<\frac{1}{2},$ $|a_{n_{2}}|<\frac{1}{2^2},\ldots.$ Therefore, \begin{align} \sum_{j=1}^N|a_{n_j}|<\sum_{j=1}^{N}\frac{1}{2^j}<1 \end{align} This implies $\left\{\sum_{j=1}^Na_{n_j}\right\}$ is absolutely convergent, hence convergent. Is there another way to prove?