Factors in Convergent Product Converge to One

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Theorem

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

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ 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 $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ has a nonzero limit.

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

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