# Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody’s answering this question so I’ll try it here. This is really a reference request: Has a certain kind of proof ever been used?

• A series $\displaystyle\sum_n a_n$ converges absolutely if $\displaystyle\sum_n |a_n|<\infty$.
• It converges unconditionally if it converges to a finite number and all of its rearrangements converge to that same number.

For series of real numbers these are equivalent.

There are many proofs of absolute convergence of particular series, and unconditional convergence follows.

My question is whether there are any known direct proofs of unconditional convergence without deducing it from absolute convergence? And are there cases where that method is preferable? Or where absolute convergence was proved by deducing it from unconditional convergence (which would then have to be proved by some other method)? Perhaps where that was the only readily available way to do it?

(Recently I thought I was close to having one of those, and possibly I was, but I decided to do it by proving absolute convergence instead.)