I tried to look if this was already answered on here. Couldn't find it, but maybe I am just not searching well.
Let $\sum_{n=1}^\infty a_n$ be a series with nonnegative terms which converges. Let $\{a_{n_k}\}$ be a subsequence of $\{a_n\}$. Prove that $\sum_{k=1}^\infty a_{n_k}$ converges and $\sum_{k=1}^\infty a_{n_k} \le \sum_{n=1}^\infty a_n$.
I tried some things, like that the series consisting of the subsequences must be bounded above, but this didn't lead anywhere.
For each $m=1,2,3,\dots$, let $S_m = \sum_{k=1}^m a_{n_k}$, and let $T_m = \sum_{n=1}^m a_n$. Note that $\lim_{m\to\infty} T_{n_m} = \sum_{n=1}^\infty a_n$. That is, the subsequence of partial sums $T_{n_1},T_{n_2},T_{n_3},\dots$ converges to $\sum_{n=1}^\infty a_n$. (Why?)
Then for each $m$, we have $0\le S_m \le T_{n_m} \le \lim_{m\to\infty} T_{n_m} = \sum_{n=1}^\infty a_n$. Since $S_m \le S_{m+1}$ for each $m$, what can you conclude about $\lim_{m\to\infty} S_m$?