Convergent Product Satisfies Cauchy Criterion

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

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

Then it satisfies Cauchy's criterion for products.

Proof
Let $\epsilon > 0$.

Let $n_0\in\N$ be such that $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $a\in\mathbb K\setminus\{0\}$.

By Convergent Sequence is Cauchy Sequence, there exists $N_0\geq n_0$ such that $\displaystyle \left\vert \prod_{n \mathop = n_0}^k a_n - \prod_{n \mathop = n_0}^l a_n\right\vert \leq\epsilon$ for $k,l\geq N_0$.

By Sequence Converges to Within Half Limit, there exists $N_1\geq n_0$ such that $\displaystyle \left\vert \prod_{n \mathop = n_0}^Ma_n \right\vert \geq \frac{|a|}2$ for $M\geq N_1$.

Let $N=\max(N_0,N_1)$.

For $N+1\leq k\leq l$:

Hence the result.

Also see

 * Uniformly Convergent Product Satisfies Uniform Cauchy Criterion