A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsOe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.