I have recently learned about Riemann's rearrangement theorem, and I have some questions regarding the theorem.
Let $$\sum a_n = a_1 + a_2 + a_3 + a_4 + \cdots$$ be conditionally convergent. If we switch two of the terms, for example $a_2$ and $a_3$, we get the new series $$a_1+a_3+a_2+a_4+\cdots.$$ Both the new and the old series have the same sum, and this will be the case no matter which two terms we switch. Then my (obviously wrong) claim is this: A rearrangment is just repeated switching of terms.
Why is my claim wrong? I can imagine that it has something to do with the fact that for many rearrangements one has to switch two terms infinitely many times. If that is the case, I guess that one can do any finite amount of switching without altering the sum. Is this correct?
I know that it was Riemann who first proved this theorem, but I can't find out when and where. Did he write a paper on it, and if he did, where can I find a copy of it?
I am also grateful for any related information that could be of interest.
Riemann's rearrangement theorem: If an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges. (Wikipedia)
I appreciate all help.