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.)

For large $L$, take $a_0 = L - 1$ and $a_n = 2^{-n}$ for $n \geq 1$. Then for any subset $I\subset \mathbb{Z}^{\geq 0}$, the sum $s = \sum_{n\in I} a_n$ has $s \geq L -1$ if $0\in I$ and $s\leq 1$ otherwise. 
