Let $\{x_n\}_{n\in\infty}$ be a sequence. Define for each $n$ the following subsets:
$S_n := \{x_i \mid i\leq n\}$ and $\text{ }$ $T_n := \{x_i\mid i\geq n\}$.
Now, using the $\textit{supremum}$ as a function on subsets, create the corresponding sequences:
$\{\sup(S_n)\}_{n\in\mathbb{N}}$ and $\{sup(T_n)\}_{n\in\mathbb{N}}$.
The question is, are the limits of these sequences the same? I.e. does
$\lim\limits_{n\to\infty}\{\sup(S_n)\}_{n\in\mathbb{N}} \text{ }=?\text{ } \lim\limits_{n\to\infty}\{\sup(T_n)\}_{n\in\mathbb{N}} \text{ }\text{ }\left(=: \limsup\{x_n\}_{n\in\mathbb{N}}\right)$
In english, this question is asking why we define limsup using the so called "tails" of the sequences instead of the "heads".
Computationally speaking, computing sup's for the heads and analyzing the trend would be more feasible than doing so for the infinite tails (as well as more intuitive from a graphical standpoint).
[Update: The "=?" does not hold as has been pointed out. The easy counter example being $x_n := 1/n$. It looks like monotone decreasing sequences would yield counter examples in general. The ones I tried were all increasing or oscillatory of constant amplitude (e.g. $x_n := Cos(n*\pi/2)$) or both. Thanks everyone!]
