Suppose $(X_n,d_n) \rightarrow (X_{\infty}, d_{\infty})$ in the Gromov-Hausdorff sense. Is there a compact (or locally compact) space $Z$ such that (almost all) the $X_i$ embed isometrically into $Z$ and and $d_H(X_n, X_\infty) \rightarrow 0$ where $d_H$ is the Hausdorff distance in $Z$.
If not, is this true when $X_n$ is a smooth manifold, and $d_n$ is a distance inducing the topology (if not, then $d_n$ a Riemannian distance)?