  We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a
noncommutative analogue of the Gromov compactness theorem for the
Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz
compact quantum metric spaces of the closure of finite dimensional
quasi-Leibniz compact quantum metric spaces for the dual propinquity. While
finding characterizations of this class proves delicate, we show that all
nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits
of finite dimensional quasi-Leibniz compact quantum metric spaces. This result
involves a mild extension of the definition of the dual propinquity to
quasi-Leibniz compact quantum metric spaces, which is presented in the first
part of this paper.
