Factors in Convergent Product Converge to One

Theorem
Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ be convergent.

Then $a_n\to0$.

Proof
By definition of convergent product, there exists $n_0\in\N$ such that $a_n\neq-1$ for $n\geq n_0$ and the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty \left({1 + a_n}\right)$ has a nonzero limit.

Let $p_n$ denote the $n$th partial product of $\displaystyle \prod_{n \mathop = n_0}^\infty \left({1 + a_n}\right)$.

For $n> n_0$, $1+a_n = \frac{p_{n}}{p_{n-1}}$.

By the Combination Theorem for Sequences, $1+a_n\to1$.

Also see

 * Factors in Absolutely Convergent Product Converge to One