Rearrangement theorem for absolutely convergent series

Rearrangement theorem (absolute convergence): If $\sum_{n=1}^\infty a_n$ converges absolutely in $\mathbb{R}$ or $\mathbb{C}$ and $\pi:\mathbb{N}\to\mathbb{N}$ is a bijection, then the rearranged series $\sum_{n=1}^\infty a_{\pi(n)}$ converges, and $\sum_{n=1}^\infty a_{\pi(n)}=\sum_{n=1}^\infty a_n.$

Absolute convergence guarantees stability of infinite sums under reindexing, a property that fails for conditionally convergent series .

Proof sketch (optional): Given $\varepsilon>0$, choose $N$ so that the tail $\sum_{n>N}|a_n|<\varepsilon$. Any partial sum of the rearranged series that contains all indices $\le N$ differs from the full sum by at most $\varepsilon$ in absolute value, forcing convergence to the same limit.