On the level convergence of a sequence of fuzzy numbers

In this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space E1 established by Wu and Wu (J. Math. Anal. Appl. 210 (1997) 499) and improve the expressions of the supremum and infimum. As a straightforward corollary of this result, we obtain a necessary and sufficient condition under which, for a bounded sequence of fuzzy numbers {un}, the pair of functions supnun-((lambda)) and supnun+((lambda)) can determine a fuzzy number. Secondly, we give a necessary and sufficient condition for a sequence of fuzzy numbers {un} to be levelwise convergent in E1, and generalize some important theorems in real number spaces to fuzzy number spaces. Finally, we prove the existence of supremum and infimum for the level-continuous fuzzy-valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained.