Absolute convergence implies the Cauchy criterion

Corollary: Let $(a_n)$ be a real or complex sequence. If $\sum_{n=1}^\infty |a_n|$ converges , then for every $\varepsilon>0$ there exists $N$ such that for all $m>n\ge N$, $\sum_{k=n+1}^{m} |a_k| < \varepsilon.$ Consequently, the partial sums of $\sum a_n$ form a Cauchy sequence , so $\sum a_n$ converges.

Connection to parent theorem: This is the Cauchy criterion applied to the convergent series of nonnegative terms $\sum |a_n|$, combined with the estimate $\left|\sum_{k=n+1}^m a_k\right|\le \sum_{k=n+1}^m |a_k|.$