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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)?

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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?

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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.

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