Eternal the originator of time - proof.

[Mystic]

A chain of effects is an effect.
An effect requires a cause.
A chain of effects thus requires a cause.

This is a logically sound argument. However, when we talk about infinity, there is always a prior chain of effects that can act as a cause. Wrestling with infinities, we can often construct scenarios which end in apparent contradictions (such as your infinite army chain who shoot when the next person in line tells them to - how could it possibly start?). The contradictions in question are not necessarily logical contradictions. They can simply be counter-intuitive conclusions, that are nevertheless mathematically provable.

Infinite chain of effects is still an effect. So it would by definition require a cause but at the same time due to it's infinite nature cannot be caused so then a paradox, thus being impossible.