  We consider the Lion and Man game, i.e., a two-person pursuit-evasion game
with equal players' top speeds. We assume that capture radius is positive and
chosen in advance. The main aim of the paper is describing pursuer's winning
strategies in general compact metric spaces that are close to the given one in
the sense of Gromov-Hausdorff distance. We prove that the existence of
$\alpha$-capture by a time $T$ in one compact geodesic space implies the
existence of $(\alpha + (20T +8)\sqrt{\varepsilon})$-capture by this time $T$
in any compact geodesic space that is $\varepsilon$-close to the given space.
It means that capture radii (in a nearby spaces) tends to the given one as the
distance between spaces tends to zero. Thus, this result justifies calculations
on graphs instead of complicated spaces.
