in my book this is called the lower bound but it implies that it might be called the greatest lower bound elsewhere. lower bound: some quantity m such that no member of a set is less than m but there is always one less than m + [itex]\epsilon[/itex] definition using Dedekind section there are quantities a such that no member of a set is less than any a. there are quantities b such that some members of a set are less than any b. all a(s) are less than all b(s)[anyone have a problem with this???]. and all quantities of the same dimension are either a(s) or b(s). a(s) define the L class and b(s) define the R class. now make a partition, m, that has to be in L. for if it were in R then there would be some member of the set, k, less than it and and there would be no a(s) between m and k and hence it wouldn't partition correctly. m is then the greatest lower bound. m + [itex]\epsilon[/itex] is in R which by prior definition must have a member less than it. the part in brackets i'm not sure about specifically besides not being sure about the whole thing, despite the fact that i basically followed the Dedekind proof/defition for least upper bound. this is the first mathematical proof by argument i have ever done and boy was it more abstract than any physics problem i've ever done.