Let $(a_n)$ be a real sequence and consider the series $\sum_{n=1}^\infty a_n$. I want to show that the series converges if and only if the following property holds: $\forall \epsilon>0, \exists N\in \mathbb{N} \ s.t. \forall n>N, |\sum_{k=n}^\infty a_n|<\epsilon$.
(I know that the series is convergent if and only if it satisfies the Cauchy criterion: $\forall \epsilon>0, \exists N\in\mathbb{N}\ s.t. \forall n\geq m>N, |\sum_{k=m}^n a_n|<\epsilon$.)
To show that if the series converges, then the property above holds, We first suppose that $\sum_{n=1}^\infty a_n$ converges. Then, since $\sum_{n=1}^\infty a_n = \lim s_n$ where $(s_n)$ is the sequence of partial sums, then by definition of a limit of a sequence, $\forall \epsilon>0, \exists N\in\mathbb{N} \ s.t. \forall n>N, |\sum_{k=n}^\infty a_n|=|\lim_{k\rightarrow \infty} (s_k-s_{n-1})|=|(\lim_{k\rightarrow \infty}s_k) - s_{n-1}|<\epsilon$.
However, how do I show the converse? When the convergence of the series is unknown, how can we write down $\sum_{k=n}^\infty a_n$?