= Vanishing of infinitely divisible integer sequences
{title2=$\bigcap_{i\geq0}2^i\prod_{j\in J}\mathbb Z=\{0\}$}
If an element $w$ of the <module> $\prod_{j\in J}\mathbb Z$ is divisible by every power of two within that <module>, then every coordinate $w(j)$ is an <integer> divisible by every power of two. A nonzero <integer> $a$ cannot be divisible by $2^i>|a|$, so each coordinate is zero and $w=0$. This argument applies to any index set $J$ and to powers of any fixed integer $b$ with $|b|>1$.
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