Convergence of series from convergence of all sparse sub-series? Suppose $\sum_n a_n$ is a series of positive decreasing terms. Suppose that for each possible sub-series $\sum_k a_{n_k}$ where $n_1 < n_2 < \ldots$ we have that the sub-series converges if $\lim_{k \to \infty} k/n_k = 0$.  Must the original series $\sum_n a_n$ converge?
 A: I was hoping for some cool direct proof showing some convergence criterion holds, but I have a proof by contrapositive. Assume a series $\sum_n a_n$ of positive decreasing terms diverges. Then every arithmetic progression summation of term indices $\sum_k a_{j + mk}$ diverges for every starting index $j \geq 1$ and every increment integer $m \geq 1$. Thus we can construct a divergent sub-series as follows. In the first iteration, first choose terms in order starting from the beginning until the sum is greater than $1$. Then in the second iteration, choose every second term starting after the last term chosen in the first iteration, until the sum of terms chosen in the second iteration is greater than $1$ and also such that the number of chosen terms is more than twice the number of terms chosen in the first iteration. And so on, in the third iteration choose every third term starting after the last term chosen from the second iteration, until the sum is greater than $1$ and the number of terms chosen is twice the number of terms chosen in the first two iterations. The sub-series so constructed will diverge, and also satisfy $\lim_{k\to \infty} k/n_k = 0$.
