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.