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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.

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  • $\begingroup$ 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? $\endgroup$ – Kevin Oct 29 '15 at 11:00
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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.

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