  We investigate extensions of S. Solecki's theorem on closing off finite
partial isometries of metric spaces \cite{solecki1} and obtain the following
exact equivalence: any action of a discrete group $\Gamma$ by isometries of a
metric space is finitely approximable if and only if any product of finitely
generated subgroups of $\Gamma$ is closed in the profinite topology on
$\Gamma$.
