Cauchy's Convergence Criterion for Series

Theorem
A series $\displaystyle \sum_{i \mathop = 0}^\infty a_i$ is convergent for every $\epsilon > 0$ there is a number $N \in \N$ such that:


 * $\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$

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

Proof
Let:


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

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

From Cauchy's Convergence Criterion, $\sequence {s_n}$ is convergent it is a Cauchy sequence.

For $m > n$ we have:


 * $\size {s_m - s_n} = \size {a_{n + 1} + a_{n + 2} + \cdots + a_m}$