\begin{definition}[Definition:Cauchy's Criterion for Products]
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''' {{Iff}}:
:$\ds \forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m\geq n \ge N: \norm {\prod_{k \mathop = n}^m a_n - 1} < \epsilon$
\end{definition}