The completion theorem in K-theory for proper actions of a discrete group

We prove a version of the Atiyah–Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K∗(EG×GX) is the completion of KG∗(X) with respect to a certain ideal. We also show, for such G and X, that KG(X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X.