Set convergence and lim inf and lim sup

Question

I'm a bit confused with the general concept of convergence of a sequence of sets.

I'm well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = \limsup_{\nu \rightarrow \infty} C^{\nu}$$ where lim inf (resp. lim sup) is the set of points that appear in the limit all but finitely many times (resp. infinitely many times).

However, intuitively, the limit point can appear only once, i.e., for $\nu \rightarrow \infty$. Isn't this in contrast with the concepts of lim inf and lim sup (defined as above)?

For instance, let $C^{\nu} \triangleq [0,1-1/\nu]$: the sequence $\{C^{\nu}\}$ should (intuitively) converge to $C \rightarrow [0,1]$. However, I think the point $\{1\}$ is included in $C$ only for $\nu=\infty$ and, therefore, it appears only once.

What am I missing?

Answer 1

There are two competing definitions of $\liminf$ and $\limsup$. The one referenced by @V.C. in the comments is this: $$\liminf_{\nu\to\infty} C_\nu := \bigcap_{n\in\mathbb{N}}\bigcup_{\nu\ge n}C_\nu$$ which we can rewrite in the notation of Rock & Wets as $$\liminf_{\nu\to\infty} C_\nu := \bigcap_{N\in\mathcal{N}_\infty^\#}\bigcup_{\nu\in N}C_\nu.$$ The definition from Rock & Wets, and other sources such as Hocking & Young, adds a closure: $$\liminf_{\nu\to\infty} C_\nu := \bigcap_{N\in\mathcal{N}_\infty^\#} \text{cl}\bigcup_{\nu\in N}C_\nu.$$ (For $\limsup$ simply transpose the union and intersection operators.)

For what it's worth, Wikipedia uses the latter definition, written equivalently as:

lim inf Xn, which is also called the inner limit, consists of those elements which are limits of points in Xn for all but finitely many n (i.e., cofinitely many n). That is, x ∈ lim inf Xn if and only if there exists a sequence of points {xk} such that xk ∈ Xk and xk → x as k → ∞.

Note that in a discrete space the two definitions coincide. $\mathbb{R}^n$ with the usual topology is not a discrete space.