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While proving a theorem, the Professor stated the following lemma without proof: Let $a_n$ be a non increasing positive series, such that $\sum_n a_n=\infty$ One can find a positive series $c_n$ such that $\sum_n c_n=\infty$ and $\lim_{n\to\infty} \frac{c_n}{a_{bn}}=0$ for all integer $b>0$

I tried to use the fact that given a series as $a_n$, one can show that $\sum_n \frac{a_n}{S_n}=\infty$ when $S_n=\sum_{i=1}^{n}a_i$ But it only proves the case of $b=1$. A further attempt was to assume that $a_n<1/n$ as for $a_n=1/n$ we can choose $c_n=\frac{1}{n\log{n}}$, and also noticisng that in that case ($a_n<1/n$) $S_{n^2} \le 2S_n$ and thus we also have $\sum_n \frac{a_n}{S_{n^2}}=\infty$, but I could not not prove from here the desired limit.

user580021
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Your idea still works even when $b>1$. That's because your sequence $a_n$ consists of positive terms, thus $0\leq\sum_n \frac{a_n}{S_{bn}}\leq \sum_n \frac{a_n}{S_{n}} $. The righhand side converges to zero, so by the squeeze theorem $\sum_n \frac{a_n}{S_{bn}}$ converges to zero as well.

pirahahindu1999
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