Equivariant embedding of metrizable $G$-spaces in linear $G$-spaces

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 $M$.