The Riemann rearrangement theorem, and an interesting corollary
I reencountered the Riemann rearrangement theorem this weekend. It states that a conditionally convergent series can be rearranged so that it adds up to any given number. Mysterious-seeming, yes, but not at all hard to justify: it almost follows directly from the definition of conditional convergence. If a series 
converges conditionally, then 
and 
, so you can approximate any positive number by adding an appropriate number of terms from the positive 
followed by an appropriate number of terms from the negative ; since , this approximation can be made arbitrarily precise. Ditto for negative numbers.
Seems this is yet another one of those simple theorems that just didn’t click with me the first time around– I suspect because I was introduced to it via a formal proof, where we had to keep track of multiple quantities, instead of the simpler hand waving argument I just gave. As I recall it, the formal proof in baby Rudin uses an atrocious number of auxiliary quantities.
An interesting corollary (at least to me, since I came up with it :)) of the Riemann rearrangement theorem is an easy proof that the set of permutations on 
is uncountable: the rearrangements of any conditionally convergent sequence correspond to permutations of the natural numbers, and a rearrangement of the harmonic series can be found that sums to any real number, so there is an onto mapping from the set of permutations of the natural numbers to the real numbers. Beats diagonalization arguments any day.
Possibly relevant posts:
- An Inversion Theorem for Formal Power Series (1/2/2005)
- Approximate Identities (7/2/2005)
- Dr. Singh’s Take on the Fourier Unicity Theorem (via Invariant Subspaces) (6/17/2005)
>Beats diagonalization arguments any day.
if you want to avoid diagonalization, you should also prove that the reals are uncountable independently, i guess one can do this by saying that
it contains a perfect subset and perfect sets can be
proved to be uncountable without using iagonalization.
both of what you guys say is very interesting, but i still think that diagonalization is very elegan
The rearrangement theorem has important applications in the field of white collar crime and fraud: you can misrepresent your company simply by counting your faster than your liabilities (if you want to fool the stockholders) or vice versa (if you want to fool the IRS).