This paper is the sixth in a series designed to supplement the statistics training of students. The intended audience is social science undergraduate and graduate students studying applied statistics. The purpose of the applied statistics monographs is to provide selected proofs and derivations of important relationships or formulas that students do not find available and/or comprehensible in journals, textbooks and similar sources. Derived is the theoretical limits of the sample multivariate (or multiple) correlation of one criterion (dependent variable) and any (finite) number of predictors (independent variables). The proof given in this paper involves deriving the individual terms of R. The lower limit and upper limit of R are derived separately. (KR)