(related to Proposition: Cauchy Criterion)
Let \(s_j:=\sum_{k=0}^j x_k\) be the \(j\)-th partial sum of the infinite real series \(\sum_{k=0}^\infty x_k\) and let \((s_j)_{j\in\mathbb N}\) be the real sequence of these partial sums. The proposed condition for the convergence states that for every \(\epsilon > 0\) there is an index \(N(\epsilon)\in\mathbb N\) such that
\[\left|\sum_{k=m}^n x_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
Since
\[s_n-s_{m-1}=\sum_{k=m}^n x_k,\]
we get
\[\left|s_n-s_{m-1}\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
This means that the real sequence \((s_j)_{j\in\mathbb N}\) is a real Cauchy sequence. According to the completeness principle every real Cauchy sequence is convergent. If follows that \(\sum_{k=0}^\infty x_k\) is a convergent real series.
