Abstract
The purpose of these studies is to clarify functions of mathematical proofs in school mathematics, especially constructing processes of mathematical conception. We can find a theorem which describes a relation between two sets. In this paper, we consider such kind of theorem, which is limited to the case where the relation is a function between two sets. In this case, we can prove the theorem by constructing a function between two sets. In this constructing process, we make a family of sets and functions with an indexing set - a sequence of sets and functions. This indexing set is usually an ordered set, natural numbers, real numbers, for example. From this setting, we can construct two kinds of limits. One is a projective limit; the other is an inductive limit. We can construct a function from these structures as a function from the inductive limit or a set of a proper restriction of this, to the projective limit. Form these constructions, we can reform other constructions by embedding the indexing set such as natural numbers into other ordered set such as a positive real numbers. As a result of this reform, we can get a more suitable process of proving the theorem, a more useful method for constructing the function or a more important function between the two sets. This means that ordered sets has important role in constructing process of functions that is proof making process.
