I am trying to think of examples of a compact connected subset $S$ of Euclidean space which is the set of limits of all converging sequences $\{ a_n \} $ such that $a_n$ is in $S_n$, where the $S_n$'s are also compact connected sets which converge to the set $S$ with respect to the Hausdorff distance, are there some clear examples of this?
2026-03-25 14:27:15.1774448835
A Compact Set as the Set of Limits of all Sequences
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For instance, in a Euclidean space $\Bbb R^2$ for each $n$ we can put $S_n$ to be a closed disk of radius $1/n$ centered at the origin $(0,0)$ of the space. Then $S=\{(0,0)\}$.