Note that it's assumed the man and the lion are points with no width. The lion doesn't have to get close to win, it has to be in the exact same position. This i

Note that it's assumed the man and the lion are points with no width. The lion doesn't have to get close to win, it has to be in the exact same position. This is mentioned near the end of the post (should be at the start). If the lion only had to get close, it would clearly win by just continuously running directly towards the target. The curve it traced would cover less distance whenever the target was forced to turn, and so the lion can get arbitrarily close (assuming things like an instantaneous reaction time). The important detail, that allows the man to escape when the positions must match exactly, is that the lion's tracing-smaller-curve advantage goes down as it gets closer. It has to match the turns being made by the man more and more exactly in order to not lose ground. Work out the result, and the corresponding infinite sequence fails to converge.