Convergent Product Satisfies Cauchy Criterion

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 it satisfies Cauchy's criterion for products.

Proof
Let $\epsilon > 0$.

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

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

for $k, l \ge N_0$.

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

for $M \ge N_1$.

Let $N = \max \set {N_0, N_1}$.

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

Hence the result.

Also see

 * Uniformly Convergent Product Satisfies Uniform Cauchy Criterion