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 ( 2 n + 2 ) -point sets has a Vietoris continuous selection provided that one of at most ( 2 n + 1 ) -point sets has such a selection. This settles some open questions.