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 $\ds \prod_{n \mathop = 1}^\infty a_n$ satisfies Cauchy's criterion if and only if:

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

Also known as

Some sources refer to the Cauchy criterion as the Cauchy condition, but the same term is used for the Cauchy boundary condition.

Hence in order to reduce possible confusion, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the use of Cauchy criterion

Also see

• Convergent Product Satisfies Cauchy Criterion