Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence

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 $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ 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 $b \in \mathbb K \setminus \set 0$.

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