Abstract: The weakly almost periodic point and the periodic sequence shadowing property are studied under strongly uniform convergence. Some conclusions about them are obtained. First, let the sequence map f_n converge strongly uniformly to the equicontinuous map f and the sequence of pointsx_kbe the weakly almost periodic point of every map f_n. If \mathop \lim \limits_k \to \infty x_k = x, then the point x is the weakly almost periodic point of the map f. Second, if the sequence mapf_nconverges strongly uniformly to the equicontinuous map f, then limsup W(f_n)⊂W(f). Third, if f_n has the fine periodic sequence shadowing property, then f has periodic sequence shadowing property.