Absolute convergence implies convergence

Absolute convergence implies convergence: Let $(a_n)$ be a real or complex sequence. If the series $\sum_{n=1}^\infty |a_n|$ converges, then the series $\sum_{n=1}^\infty a_n$ converges.

Absolute convergence is stronger than convergence and guarantees many good properties (e.g., rearrangements do not change the sum ).

Proof sketch (optional): If $\sum |a_n|$ converges then its partial sums are Cauchy , implying tails $\sum_{k=m}^n |a_k|$ are small; by the triangle inequality, tails of $\sum a_n$ are also small, so $\sum a_n$ is Cauchy and hence convergent in $\mathbb{R}$ or $\mathbb{C}$.