Divergent Infinite Series Question

Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there always exist a $$\sum{a_{n_i}}<\infty$$ where obviously the sequence must have infinitely many terms.

Answer 1

Yes. Since $\lim\limits_{n\to\infty} a_n=0$, for any $i$ we have some $n_i$ such that $a_{n_i}<\frac{1}{i^2}$. Then $$\sum\limits_{i=1}^\infty a_{n_i}<\sum_{i=1}^\infty \frac{1}{i^2}=\frac{\pi^2}{6}$$