What does $n \to \infty$ suggest in the context of limit superior (or inferior)? This question is about notations and notations only and it may sound unnecessarily pedantic or nitpicky but I'm kind of curious what does $n \to ∞$ below the $\limsup $ or $\liminf $ notations suggest?
For instance, the limit superior of a sequence $(x_n)$ is denoted by writing $\limsup\limits_{n \to ∞} x_n$ instead of simply writing $\limsup x_n$ (which, I'm aware is an alternative notation).
Now, I could write this question off as "Hey that's just a notation." But what bothers me is that $\limsup x_n$ is simply the greatest limit point of the sequence $(x_n)$ whether or not $n$ is large. Then why write $n \to \infty$, I wonder?
In contrast, writing $\lim\limits_{n \to ∞} x_n = L$ means when $n$ is sufficiently large then $x_m≈L, \, \forall \,m≥n$. But I can't make a similar sense of those notations used for limit superior (or inferior). Thoughts?
 A: Summarizing the comments:

For a sequence $(x_n)$, one definition of limsup is
$$\limsup_{n \to \infty}x_n := \lim_{n \to \infty}\left(\sup_{k \geq n}x_k\right),$$
so there is in fact a limit involved. As with limits, you can drop the $n \to \infty$ when there's no ambiguity. An example where it would be ambiguous to drop it is
$$\limsup_{n \to \infty} a_{m,n}$$
where $(a_{m,n})$ is a sequence of two variables.
It is worth noting that, as with limits, the limsup/liminf concepts also apply to functions of real variables. For example,
$$\limsup_{x \to 0}f(x) := \lim_{x \to 0}\left(\sup\{f(y) : 0 < |y| < |x|\}\right)$$
Here, the $x\to 0$ cannot be omitted without ambiguity, because it tells us which point we're focusing on.
So to summarize, for sequences of one variable, it's true that the $n \to \infty$ is redundant (both for limsup/liminf and for lim) because there's no other interpretation that makes sense. I think people use the $n \to \infty$ for consistency with the other situations (sequences of multiple variables, functions of a real variable) where similar notation is required. In any case, it doesn't harm anything to be explicit even when not strictly necessary.
