Question on Riemann's rearrangement theorem Let there be given some conditionally convergent infinite series $S$.  Then let $R$ be some real number, and $Q_k$ a rearrangement of $S$ such that the sum is equal to $R$.
Is $Q_k$ unique?  In other words, is there some other rearrangement $W_k$ for which the sum of $S$ is also $R$?
I believe the answer is that $Q_k$ 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?

The line of reasoning is not valid:
If you find a particular $Q_k$, then I can find infinitely many other rearrangements that give the same value:  Pick any finite subset $X$ of $\mathbb{N}$ and let $\sigma$ be any permutation of $X$.  Then the re-rearrangement where we start with $Q_k$ and just mix up the subscripts in $X$ according to $\sigma$ will yield the same sum.
The point is that only rearranging  a finite number of terms doesn't change the sum.
Your reasoning breaks down where you say "and therefore the mapping should be 1-1".  Just because two sets have the same cardinality doesn't mean any function you think of should be 1-1.  Having the same cardinality just means "there is some very specific choice of function which is 1-1 and onto" - it says nothing at all about a generic function you pick between two sets.
