Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence

Definition
Let $\struct {\mathbb K, \size{\,\cdot\,}}$ be a valued field. Let $(a_n)$ be a sequence of elements of $\mathbb K$.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent :
 * There exists $n_0\in\N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b\in\mathbb K\setminus\{0\}$.

The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a\in\mathbb K$.