CONVERGENT SUBSEQUENCES FROM SEQUENCES OF FUNCTIONS ( i )

Let \yA be a sequence of functions, y. e TlseSE where S is a nonempty subset of the /-dimensional Euclidean space and 77 is an ordered vector space with positive cone X . If y, £"sfji,i sufficient conditions are given that \y A have a subsequence \hA such that for each t e S the sequence {A.(i)| is monotone for k sufficiendy large, depending on i. If each E is an ordered topological vector space, sufficient conditions are given that \y A has a subsequence \h,\ such that for every í eS the sequence {¿z,(i)j is either monotone for k sufficiently large depending on t, or else the sequence |zz,(t)i is convergent. If E = 73 for each s and B a Banach space then a definition of bounded variation is given so that if \y. j is uniformly norm bounded and the variation of the functions y, is uniformly bounded then there is a convergent subsequence \hA of \yA. In the case E a E fot each s eS and E is such that bounded monotone sequences converge then the given conditions imply the existence of a subsequence \h,\ of \y A which converges for each t e S.