Factors in Convergent Product Converge to One

From ProofWiki
Jump to navigation Jump to search

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 $a_n\to0$.


Proof

By definition of convergent product, there exists $n_0\in\N$ such that:

$a_n\neq0$ for $n\geq n_0$
the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ has a nonzero limit.

Let $p_n$ denote the $n$th partial product of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$.

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

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

$\blacksquare$


Also see