As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time. This is all

As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time. This is all but obvious to me. I pick a random number from 0 to 2*Pi. If you guess my number correctly, you get to eat me. You can guess an infinite amount of times. Who wins this game? Does the answer depend on the assumption of fundamental axioms?