Factors in Convergent Product Converge to One

Theorem

Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.

Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ be convergent.

Then $a_n \to 1$.

Proof

By definition of convergent product, there exists $n_0 \in \N$ such that:

$a_n \ne 0$ for $n \ge n_0$

the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ has a nonzero limit.

Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = n_0}^\infty a_n$.

For $n > n_0$:

$a_n = \dfrac {p_n} {p_{n - 1} }$

By the Combination Theorem for Sequences:

$a_n \to 1$

$\blacksquare$

Also see

• Factors in Absolutely Convergent Product Converge to One