Let $(C_n)_n$ be a sequence of nonempty subsets of $[0,1]$ (we can assume that the $A_n$ are open/closed if it helps). Let $P$ be a nonempty subset if $[0,1]$.
What is a natural way to formalize the claim "$C_n$ is eventually a subset of $P$", or, in other words, "In the limit, $C_n$ is a subset of $P$" without requiring that any $C_n$ be a subset of $P$?
If $(C_n)_n$ is decreasing, then it would be natural to simply require that $\cap_n C_n \subseteq P$. But I don't want to restrict myself to decreasing sequences.
Another idea I had was to use the Hausdorff metric to induce a topology on the space of nonempty subsets of $[0,1]$. Then we can at least talk about convergence of sequences of subsets in a rigorous way. But even after doing that, I'm not sure how to capture the idea I want. Any suggestions would be appreciated.
Since your question is rather open-ended I'll mention the following construction:
$$B = \bigcap_{n=0}^{\infty}\left( \bigcup_{i=n}^{\infty} C_i\right)$$
$B$ is the set of points that are in an infinite number of the $C_n$. Notice that it takes a possibly non-decreasing sequence of sets and converts it to a decreasing one by taking less and less of the "infinite tail". Your saying what you thought would be good for a decreasing sequence reminded me of it. And of course the "subset" part would translate to $B \subset P$.
Although the word I associate with this construction isn't "eventually" but "essential", as in "B is the essential part of $(C_n)_n$", because points that are in only a finite number of the of the $C_n$ get left out of $B$ -- you have to really be an essential member of the sequence to end up in $B$. I'm curious to know how this might correspond to your intuition.