## refinement of Riemann theorem

Please, help to prove the following refinement of Riemann theorem about rearrangements of conditionally convergent series.
Suppose that series $\sum^\infty_{n=1} u_n=s$ converges conditionally. Then for each $s'>s$ there exists a permutation of positive integers $\sigma:\mathbb{N}\to\mathbb{N}$ such that

1) if $u_n\geq 0,$ then $\sigma(n)=n;$

2) $\sum^\infty_{n=1} u_{\sigma(n)}=s'.$

The standart proof of Riemann's theorem fails because of the condition 1). The idea was to choose infinite subset $F$ of indices $n$, such that $u_n<0$ for $n\in F$ and $\sum_{n\in F} u_n>-\infty$ and then to use this set to enlarge the sum. But I don't know how to do it correctly.