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Description:
The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.