Consider the following Kantian (nevermind whether it's actually Kant's) reason for believing in eternal life: In a finite amount of time, we cannot achieve moral perfection, but moral perfection is a basic aim of ours, and basic aims of ours are achievable. Actually, it doesn't work. For if moral perfection cannot be achieved in a finite amount of time, then moral perfection cannot be achieved by us, at least not without seriously fooling with the metric structure of time (could one have a temporal structure where an infinite life is followed by a further time of life?) For at any given time, the life we have lived is only finite.
So, perhaps apart from weird metric structures on time, the only way the Kantian argument can work is if our aim not moral perfection, but asymptotic approach to moral perfection. But there are two objections to taking that to be our basic aim. The first is implausibility. "Be perfect!" is, to my mind, a very plausible moral goal. But "Approach perfection asymptotically!" seems much less compeling. Suppose one asks "Why?" In regard to "Be perfect!" the answer is easy: as long as you're imperfect, you are doing something immoral, and you have overwhelming reason not to do that. In regard to "Approach perfection!" one can give the same answer—but this answer supports not "Approach perfection asymptotically!" but "Be perfect!" (And if "Be perfect!" is impossible, then we have a refutation of ought implies can. I myself think "Be perfect!" is achievable with God's grace in this life, though rarely achieved and not required for eventual salvation.)
The second problem with "Approach perfection asymptotically!" is that it seems to be a goal that one can rationally put off to another day—and do so forever, thereby ensuring that one does not approach perfection asymptotically. Here is another way to put this. "Approach perfection asymptotically!" has very little to say about what I should do right now. I should not do anything that would set in me a character that would make asymptotic approach not likely. (It does not even follow from "Approach perfection asymptotically!" that I should at any given time try to maximize the probability of eventual asymptotic approach.)
What if, instead, we make the goal be: "Constantly improve morally!" But that goal is too weak. It is satisfied by the following life. Today, George causes pain to a bunny for one hour. Tomorrow, he does so for 3/4 of an hour. The day after tomorrow, he does this for 4/6 of an hour. The day after that he does this for 5/8 of an hour. The day after that, he does it for 6/10 of an hour. And so on. There is constant moral improvement, but no asymptotic approach. Nor will it help to combine "Approach perfection asymptotically!" with "Constantly improve morally!"
So, if I'm right, the moral perfection goal is "Be perfect!" And not just "Be perfect eventually, some day during an infinite life!" For that could always be put off. (Think of someone who would live an infinite life and whose goal was to do a pilgrimage to the Holy Land at some time or other. That goal could always be put off rationally, while acting compatibly with it.) Rather, the goal has to be "Be perfect in this finite life!" Or maybe even "Be perfect now!"
I should note that the perfection I am talking about here is moral perfection, which is the mere absence of vice, rather than the evangelical perfection that calls for, e.g., celibacy and selling all one's possessions. This evangelical perfection is a supererogatory perfection. Being viceless is not supererogatory.