In a Supplement to Rudin's "Principles of mathematical analysis" I have found an interesting, yet difficult and abstract, exercise on how to predict qualitatively what is the effect of a certain rearrangement (or permutation if you like) on a series. This is the exercise:
Let $\{P_n\}_n$ be an arbitrary rearrangement or permutation of the set of positive integers. For every positive integer N, let us define the mixing number $mix(\{P_n\}_n, N)$ to be the largest integer M for which there exist M positive integers $n_1, n_2, ... n_M$ which are alternately ≤N and >N, and such that $P_{n_1} < P_{n_2} < ... < P_{n_M}$.
In other words, if we color each positive integer red or blue according to whether it shows up before or after the N-th comma in the sequence $P_1, P_2, P_3, ...,$ then $mix(\{P_n\}_n, N)$ denotes the number of same-color blocks into which the set of all positive integers is divided. Since only N integers are colored red, this number is at most 2N+1; in particular, it is finite.
Now, show that the following conditions are equivalent:
- $mix(\{P_n\}_n, N)$ is bounded as a function of N (that is, for a fixed $\{P_n\}_n$)
- For every convergent real-valued series $\sum a_n$, the series $\sum a_{P_n}$ also converges to the same value
My attempt
First of all i convinced myself of the equivalence between the "mixing function" and the number of colourings through examples and drawings, then I expressed this relation rigorously so that i can actually use colourings instead of the function itself (which I think isn't much of help since it is not so intuitive). Finally I began proving the first direction of the equivalence by contradiction; in fact, it seems to me that by supposing that there is a convergent series such that its rearranged (through $\{P_n\}_n$) is divergent we can find some kind of "Cauchy Criterion attack" in order to reach an absurd. However I haven't been able to implement the hypothesis, I just noticed that it allows us to assume something in the line of "there exists a fixed point such that for all N, all the positive integers must be blue after it".
This time I would prefer an hint that gives me the sense of such a proof, but I highly appreciate any help!