Let X be a real valued random variable with E|X|r+δ < ∞ for some positive integer r and real number, δ, 0 < δ ≤ r, and let {X, X1, X2, …} be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost all w ∈ Ω, with probability 1. if for some , where 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.