Please, help to prove the following refinement of Riemann theorem about rearrangements of conditionally convergent series.
Suppose that series converges conditionally. Then for each there exists a permutation of positive integers such that
1) if then
2)
The standart proof of Riemann's theorem fails because of the condition 1). The idea was to choose infinite subset of indices , such that for and and then to use this set to enlarge the sum. But I don't know how to do it correctly.