Does every element of $ \Pi_{n=1}^\infty{\mathbb{Z}_i} $ have finite order?

Question

Does every element of $ \Pi_{n=1}^\infty{\mathbb{Z}_i} $ have finite order?

If I took an infinite direct product of $\mathbb{Z}_i$ for $i=1,2,3,\ldots.$ Would the element $\left(1,1,1,\ldots \right)$ have finite order?

Answer 1

No, of course that element can't have finite order.

For any finite number $n\ge 3$, we can prove that $n$ is not the order of your element: To wit, the $\mathbb Z_{n-1}$ component of $n\cdot(1,1,1,\ldots)$ is $1$ rather than $0$.

The order clearly also is neither $1$ nor $2$.