  Given a Lie group $G$ we study the class $\M$ of proper metrizable $G$-spaces
with metrizable orbit spaces, and show that any $G$-space $X \in \M$ admits a
closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear
$G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the
class $\M$. As corollaries we see that any $G$-ANE for $\M$ has the
$G$-homotopy type of some $G$-CW complex and that any $G$-ANR for $\M$ is a
$G$-ANE for $\M$.
