For a series, can you always find a subseries whose sum is smaller in magnitude? Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, when can you?
(Inspired by this comment.)

We cannot always do it. Look for example at the series $\sum_0^\infty \frac{1}{10^n}$.
