Riemann rearrangement theorem

Riemann rearrangement theorem: Let $\sum_{n=1}^\infty a_n$ be a conditionally convergent series of real numbers (i.e., it converges but not absolutely). Then for any $L\in\mathbb{R}$ there exists a rearrangement $\sum a_{\pi(n)}$ that converges to $L$. There also exist rearrangements that diverge to $+\infty$, to $-\infty$, and that oscillate.

This theorem highlights that conditional convergence is fragile: the “sum” depends on the order of terms.

Proof sketch (optional): Separate positive and negative terms: $\sum a_n^+$ diverges to $+\infty$ and $\sum a_n^-$ diverges to $-\infty$. Build a rearrangement by adding enough positive terms to exceed $L$, then enough negative terms to drop below $L$, and repeat with ever smaller overshoots since $a_n\to 0$.