In Durrett's Probability (4th edition), an example of a tail event (an event in the tail sigma-field $\bigcap_n \sigma(X_n, X_{n+1}, \dots)$) is the following: given independent random variables $X_1, X_2, \dots,$ and their partial sums $S_n = \sum_{i=1}^n X_i$, the following event is a tail event (Example 2.5.2):
$$ \{ \limsup_n S_n > x c_n \}, \; c_n \to \infty. $$
I understand the high level idea of a tail event (i.e. only depends in the asymptotic behavior of the sum since $c_n$ go to infinity) but I cannot articulate a rigorous explanation. Is there a concrete way to show this?
It's pretty much just like your intuitive explanation. For a any $n$, we can write $$ \frac{S_{n+k}}{c_{n+k}} = \frac{X_1+\ldots+X_n}{c_{n+k}} + \frac{X_{n+1}+\ldots +X_{n+k}}{c_{n+k}} $$ Then, since the first term converges to zero as $k\to\infty$, we have $$ \limsup_m \frac{S_m}{c_m} = \limsup_{k} \frac{S_{n+k}}{c_{n+k}} = \limsup_k \frac{X_{n+1}+\ldots+X_{n+k}}{c_{n+k}} $$ so is clearly $\sigma(X_{n+1},\ldots)$-measurable.