Let X be a real valued random variable with E|X|r+δ< for some positive integer r and real number, δ, 0<δr, and let {X,X1,X2,} be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost all wΩ, μr;n*(w)μr with probability 1. if limninfm(n)nβ>0 for some β>rδr+δ, where μr;n* is the bootstrap rth sample moment of the bootstrap sample some with sample size m(n) from the data set {X,X1,,Xn} and μr is the rth moment of X. The results obtained here not only improve on those of Athreya [3] but also the proof is more elementary.