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
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.