Let I = [ 0 , 1 ] and f n be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by F ( I ) the set of fuzzy numbers on I, and denote by ( F ( I ) , f ^ ) and ( F ( I ) , f ^ n ) the Zadeh ′ s extensions of ( I , f ) and ( I , f n ) , respectively. In this paper, we study the ω -limit sets of ( F ( I ) , f ^ n ) and show that, if all periodic points of f are fixed points, then ω ( A , f ^ n ) ⊂ F ( f ^ ) for any A ∈ F ( I ) , where ω ( A , f ^ n ) is the ω -limit set of A under ( F ( I ) , f ^ n ) and F ( f ^ ) = { A ∈ F ( I ) : f ^ ( A ) = A } .