  The pursuit-evasion game with two persons is considered. Both players are
moving in a metric space, have equal maximum speeds and complete information
about the location of each other. We study the sufficient conditions for a
capture (with a positive capture radius). We assume that Lion wins if he
manages the capture independently of the initial positions of the players and
the evader's strategy. We prove that the discrete-time simple pursuit strategy
is a Lion's winning strategy in a compact geodesic space satisfying the
betweenness property. In particular, it means that Lion wins in compact
CAT(0)-spaces, Ptolemy spaces, Buseman convex spaces or any geodesic space with
convex metric. We also do not need to use such properties as finite dimension,
smoothness, boundary regularity or contractibility of the loops.
