Definition:Convergent Product/Number Field/Arbitrary Sequence

Definition
Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$. Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$.

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

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