Selections and hyperspaces of finite sets

It is demonstrated that the hyperspace of at most (n+1)-point sets has a Vietoris continuous selection if both the hyperspace of at most n-point sets and that of exactly (n+1)-point sets have Vietoris continuous selections. This result is applied to demonstrate that the hyperspace of at most (2n+2)-point sets has a Vietoris continuous selection provided that one of at most (2n+1)-point sets has such a selection. This settles some open questions.