Existence is indeed evidence of immortality if the past is infinite

This study is concerned with the question if existence is evidence of eternal recurrence, that a current observer is within a cyclic world, if the past is infinite. Michael Huemer proposed a proof of existence being evidence of immortality using a Bayesian approach, which is discussed, as well as various counter arguments. This study then uses transition systems, a non-Bayesian approach, to prove various results about worlds that can be described by them. It is proved that in transition systems with an infinite past and a finite state set, where time can be discretely subdivided, eternal recurrence is the case for every observer in a world described by such a system. Finally, the reasoning, potential and actual counter arguments, consequences, and future research are considered.