Given a sequence of i.i.d random variables $X_n$, is the event $\{\lim_{n \to \infty} X_n \ exists\}$ in the tail algebra?
Intuitively, this should be true because the limit only depends on the tail. However, how do one prove this formally? I rewrite the event into the form $\cup_{c \in \mathbb R}\{\lim_n{X_n} = c\}$,and it seems not hard to prove that $\{\lim_{n}X_n = c\}$ is in the tail event, but it is an uncountable union.
update: I should probably reformulate the event using the Cauchy Criterion so that it is no longer an uncountable union of things. But I am not sure whether that works or not.