Cauchy's Convergence Criterion for Series

Theorem
A series $\sum_{i=0}^\infty a_i$ is convergent if and only if for every $\varepsilon>0$ there is a number $N\in\N$ such that


 * $\displaystyle\left\vert a_{n+1}+a_{n+2}+\cdots+a_{m}\right\vert<\varepsilon$

holds for all $n\ge N$ and $m > n$.

Proof
Let


 * $\displaystyle s_n = \sum_{i=0}^n a_i.$

Then $\sequence {s_n}$ is a sequence in $\R$.

From Cauchy's Convergence Criterion, $\sequence {s_n}$ is convergent if and only if it is a Cauchy sequence. For $m>n$, we have


 * $\displaystyle \left\vert s_m-s_n \right\vert = \left\vert a_{n+1}+a_{n+2}+\cdots+a_{m}\right\vert.$