Question: What's the use of talking about the $N$-tail of a sequence when it seems like all this really means is a subsequence with all of the terms from the original sequence with index greater than $N$?
Is it not true that every tail is a subsequence but not every subsequence is a tail? What do we gain from introducing the "tail" terminology other than visualizing a certain kind of subsequence? For instance, the Cauchy criterion for series is nicely visualized by recognizing that no finite number of terms effects the convergence of the series; that is, the tail of the series determines convergence or divergence.
This is a nice use for the term tail (i.e., in terms of facilitating a visualization), but are there many other uses where this is the case? I just find it odd that many analysis books (e.g., Baby Rudin) never use the term "tail" while some others do (e.g., Pugh).
Are there many other contexts where it's much more convenient to think about tails than just subsequences or vice-versa? I'm trying to better understand the motivation for such a term and some ways its use may ease the visualization of some hallmark analytical theorems perhaps.
I imagine tail to be exactly what you might think it is, the end. The "tail-end", etc. The point of this terminology as far as I have seen is that most sequences considered seemingly are the convergent ones. Therefore the points become denser toward the end, usually in "intro" analysis texts. Subsequences can be any subset of the original sequence, including the whole original sequence. Therefore since the tail or end of a sequence has points that get infinitely close to one another, a subsequence will inherit this same feature.