Fuzzy and multi-fuzzy sequences of fuzzy numbers and their convergence at a given level

This paper explores the behavior and convergence properties of fuzzy and multi-fuzzy sequences of fuzzy numbers, with a focus on their α-level convergence, α-cut boundedness, and algebraic operations. We extend the concept to multi-fuzzy sequences of dimension r, where each component sequence consists of fuzzy numbers. Through a series of illustrative examples, we analyze convergence behavior under varying α-levels, highlighting cases where convergence occurs only at α=1 or fails altogether. Furthermore, we demonstrate that the sum, difference, and product of two convergent multi-fuzzy sequences also converge under the same α-level. The paper further analyzes sequences of trapezoidal fuzzy numbers—both normalized and non-normalized—identifying necessary conditions for convergence and presenting counterexamples where convergence fails, even when sufficient conditions are true. These results contribute to a deeper understanding of fuzzy sequence behavior and set a foundation for further studies in multi-fuzzy sequence analysis and multi-fuzzy calculus.