(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.
