(As asked, I make my comment into an answer, completing it with the OP's comment) 

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.

**Added:** for clarity, I stress here the comment of Chris Gerig to Anton Petrunin's answer: Lott-Villani and Sturm's prove that the result holds even if the limiting manifold is of different dimension than the converging manifolds.