Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence

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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 $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if:

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 $a \in \mathbb K$.