Cluster Points of Subsequences

In the preceding paper [ l ] 2 Buck defines a class of "subsequences" of a multiple sequence and shows that "almost all" of such subsequences have certain properties. This note is essentially based on a different choice of the definition of "subsequences" ; that is, this paper and [ l ] are generalizations in different directions of a preceding paper by Buck and Pollard (reference 2 of [l ]). In this discussion countability is the important property of the index systems such as the integers underlying the simple sequences or the ^-tuples of integers underlying the multiple sequences. Countability is a slightly stronger condition than is necessary since the results will be shown to hold as well for functions of n variables as for multiple sequences ; some other special cases are mentioned at the end of this paper. Also I modify Buck's approach by considering cluster points in neighborhood spaces rather than limit points in convergence spaces [3]. It may be mentioned that even for multiple sequences Theorems 1 and 2 of these papers are independent since Buck's set of "subsequences" is a set of measure zero in the set of "subsequences" considered here; my Theorem 3 contains the corresponding theorem of [ l ] as a special case. Lemma 1 and its corollary, Lemma 3, are the fundamental results on which the theorems rest ; Lemma 3 is the generalization appropriate to this paper of the lemma in §3 of [ l ] .