My question is related to this question. I'm trying to apply Berge's maximum theorem. I use notation from the linked source for the rest of the question. In my case, the parameter set $\Theta$ which indexes each optimization problem is the set of natural numbers. For each $n \in \mathbb{N}$, my feasible set $C(n)$ is a compact convex subset of a Euclidean space. My question is regarding the last line: "If $C$ is both upper and lower hemicontinuous ... " Specifically, just like we know that a function defined on the set of natural numbers is always continuous, can we say something similar for correspondences, that they are always uhc and/or lhc?
More broadly, what would be a sequential analog of the statement of the Berge's theorem, if any?