Vanishing of infinitely divisible integer sequences (${\bigcap }_{i\ge 0}{2}^i{\prod }_{j\in J}\mathbb{Z}=\{0\}$)

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 $|b|>1$.