Then infinite must mean that a future and a past must both exist in infinite proportion.
If something were to cease to exist after having an infinite time before it ceased to exist then it could not cease to exist.
Because if something were to cease to exist then there could be no relative point (being the cease of existence) to the time before that occurrence. Otherwise the time before that would be finite. There are no distinguishings between relative points in an infinite, being that of a loop. Either all true or all false, therefore implying symmetry. Any piece of information that is relative, must come from a source and be taken to a source (in respect to time), there must be a beginning and an end.
If only one of the two were to exist, like in this paradox being that of only the end, then the existence would be undefined because information would not have a cause but the effect is linear. If only the beginning existed and there was no end then is would be without cause because it would be defined by the beginning but to no means of its end. It would have to be non relational, because relation provides reason, and potential to another relative point.
So I would think of this paradox as being irrational, impossible.
An infinite will always form a loop otherwise it can't be infinite.
If one thinks otherwise then tel me, lol.