I want to prove the following statement
If $\sum_{n\in I} a_n$ converges with any rearrangements of a countable index set $I$, then $\sum_{n\in I} a_n$ is absolutely convergent.
The finite case is trivial. The $\mathbb{N}$ case can be proved by contradiction invoking Riemann series theorem.
I am looking for a direct proof (hence not using contradiction).