Prove that the subsequence of a convergent series with nonnegative terms converges Let $\sum_{k=1}^\infty a_k$ be a convergent series with nonnegative terms and let $a_{n_k}$ be a subsequence of {$a_k$}. Prove that the series $\sum_{k=1}^\infty a_{n_k}$ converges.
I am having trouble figuring out how to prove this and am pretty lost. 
My very first idea was to try and prove that it is increasing but don't know where I would take it from there.
Any help would be appreciated!

Your idea is good. Let $S_l = \sum_{k=1}^l a_l$ be the sequence of partial sums of the original series and let $T_l = \sum_{k=1}^l a_{n_k}$ be the sequence of partial sums of your series. You know that $\sum_{k=1}^{\infty} a_k$ converges so let us denote the limit by $S = \sum_{k=1}^{\infty} a_k =\lim_{l \to \infty} S_l$. Since $a_l \geq 0$ both $S_l$ and $T_l$ are increasing and we have
$$ T_l = \sum_{k=1}^l a_{n_k} \leq \sum_{k=1}^{n_l} a_k =S_{n_l} \leq S $$
so $(T_l)_{l=1}^{\infty}$ is both increasing and bounded, hence tends to a limit and so $\sum_{k=1}^{\infty} a_{n_k}$ converges.
