Definition:Cauchy's Criterion for Products

Definition
Let $\struct {\mathbb K, \norm {\,\cdot\,}}$ be a valued field.

Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ satisfies Cauchy's criterion :


 * $\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m\geq n \ge N: \norm {\prod_{k \mathop = n}^m a_n - 1} < \epsilon$

Also see

 * Convergent Product Satisfies Cauchy Criterion