# If $\sum_n^\infty a_n$ diverges such that $a_n$ converges to 0. Prove that there exists a subsequence of $a_n$ such that $\sum_k a_{n_k}$ converges.

Let $\sum_n^\infty a_n$ be a divergent series of nonnegative terms such that $a_n$ converges to 0. Prove that there exists a subsequence $\{a_{n_k}\}$ of $\{a_n\}$ such that $\sum_k a_{n_k}$ converges.

• I started by: $\sum_n a_n$ converges if $\{S_{n_k}\}$ converges where this is the partial sum of $\sum_k a_{n_k}$. Need to show that $\{S_{n_k}\}$ converges. How can I do that? – Kevin Oct 29 '15 at 11:00

For any $n$, there exist a least $k_n$ such that $a_{k_n}<({1\over2})^n$.
Further choose $k_{n+1}>k_{n}$ for every $n$. Then the sequence $\{a_{k_n}\}$ has a sum $\leq 2$ and thus the series $\sum\limits_na_{k_n}$ converges.