A real infinite series \(\sum_{k=0}^\infty x_k\) is a convergent real series, if and only if the sequence of its partial sums is a Cauchy sequence, i.e. for every \(\epsilon > 0\) there is an index \(N(\epsilon)\in\mathbb N\) such that
\[\left|\sum_{k=m}^n x_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
Proofs: 1
