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 $\ds \prod_{n \mathop = 1}^\infty a_n$ 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:
- $a_n \to 1$
$\blacksquare$