Let there be given some conditionally convergent infinite series S. Then let R be some real number, and Qk a rearrangement of S such that the sum is equal to R.
Is Qk unique? In other words, is there some other rearrangement Wk for which the sum of S is also R?
I believe the answer is that Qk is in fact unique, because the cardinality of the set of permutations of the natural numbers (i.e. the positions of each element of the series S), is the same as the cardinality of the real numbers; and therefore the mapping from the real numbers to the rearrangements should be 1-1. Is this line of reasoning valid, or is this result well known?