Let's change the rules slightly. Take a ring one mile in diameter and assume the lion needs to get within 1 foot of the tamer. They both move at 1 inch per tic

Let's change the rules slightly. Take a ring one mile in diameter and assume the lion needs to get within 1 foot of the tamer. They both move at 1 inch per tick of game clock in any direction. To make things simple the the lion goes to the center of the ring, then aims for 2 inch closer to the ring than the tamer. Well the tamer will either go closer to the wall or the lion get's closer to him every second. But while he runs along the wall the lion takes a shorter path closer to the center of the ring, catches him and eats him. Also, if the tamer ever goes closer to the center of the ring in any turn the lion get's closer to him that turn because the lion stays on a line between the center of the ring and the tamer. The reason my example does not work for this puzzle is simply there is no distance closer the the center of the ring the lion can aim for and still get to eat him. At which point you need to think about what space looks like for points that are really close to each other.