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?
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.