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?