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

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

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

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