Cauchy Criterion for Series

$\sum a_{n}$ converges iff $\forall ε > 0 \exists N \in N$ such that for all $n > m \geq N$ then:

| \sum_{k = m + 1}^{n} a_{k} | = \underset{s_{n} - s_{m}}{\underset{⏟}{| a_{m + 1} + a_{m + 2} + \dots + a_{n} |}} < ε

Proof

Observe that:

| s_{n} - s_{m} | = | a_{m + 1} + \dots + a_{n} |

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