Convergent Product Satisfies Cauchy Criterion

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 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 $a \in \mathbb K \setminus \left\{ {0}\right\}$.

By Convergent Sequence is Cauchy Sequence, there exists $N_0 \ge n_0$ such that:
 * $\displaystyle \norm{\prod_{n \mathop = n_0}^k a_n - \prod_{n \mathop = n_0}^l a_n} \le \epsilon$

for $k, l \ge N_0$.

By Sequence Converges to Within Half Limit, there exists $N_1 \ge n_0$ such that:
 * $\displaystyle \norm{\prod_{n \mathop = n_0}^M a_n} \ge \frac {\norm{a}}2$

for $M \ge N_1$.

Let $N = \max \left({N_0, N_1}\right)$.

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

Hence the result.

Also see

 * Uniformly Convergent Product Satisfies Uniform Cauchy Criterion