Sum that converges to arbitrary $s$

I am just curious about this problem: suppose I know that $\sum_{n=1}^{\infty} a_n = L$ and $a_n>0$ for all $n$. Does there exist a subsequence $\{a_{n_k}\}$ such that $\sum_{k=1}^{\infty}a_{n_k}=x$ for any $0<x<L$?

I think I can let $n_1$ be the last index such that $a_1+\ldots+a_{n_1}<x$. Then I am thinking about slowly approaching $x$ using $a_{n_2}, a_{n_3}, \ldots$. However, I don't know how to formalize this or whether this is the right approach. Any hints?