Reciprocal of Absolutely Convergent Product is Absolutely Convergent

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

Let $(1+a_n)$ be a sequence of nonzero elements of $\mathbb K$.

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ converge absolutely to $a\in\mathbb K\setminus\{0\}$.

Then $\displaystyle \prod_{n \mathop = 1}^\infty \frac1{1 + a_n}$ converges absolutely to $1/a$.