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!

HINT Remember that the limit of the series is defined as the limit of the sequence of partial sums. In the case where the series only has non-negative terms, the sequence of partial sums is monotonically increasing. If you take away terms, you can only decrease the value of the sum and it's still monotonic. Now remember something about monotone and bounded sequences....
