Abstract: Let X_1, X_2, \cdots, X_n, \cdots be an i.i.d., h\left(X_1, X_2\right) a Borelmeasurable symmetroi function, E h\left(X_1, X_2\right)=\theta. The U-statistic is defined as U_n=\binomn2^-1 \sum_i=1 h\left(X_i, X_j\right) Let F_n be the empirical distribution function of X_1, \cdots, X_n, Y_1, Y_9, \cdots, Y_n be an i. i. d. sample from F_n, the bootstrap method is to approximate P\left\\sqrtn\left(U_n-\theta\right)< x\right\ by P^*\left\\sqrtn\left(U_n^*-\theta^*\right) \leqslant x\right\ as n \rightarrow \infty, where U_n^*=\binomn2^-1 \sum_i=1 h\left(Y_i, Y_j\right), \theta^*=n^-2 \sum_1 \in< < n h\left(X_i, X_j\right) and P^* denotes probability under F_n. In this note, noder E h^2\left(X_1, X_3\right)< \infty, E\left|h\left(X_1, X_3\right)\right|< \infty, the following result is oblained: \sup _g\left|P\left\\sqrtn\left(U_n-\theta\right) \leqslant x\right\-P^*\left\\sqrtn\left(U_n^*-\theta^*\right)< x\right\\right| \rightarrow 0 \quad \text a.s. Thus, the result obtained by Biokel and Freedman in 1981 is improved.