from the article: > We split time into a sequence of intervals, of lengths t1, t2, t3, . . .. At the ith step the man runs for time ti in a straight line that

from the article: > We split time into a sequence of intervals, of lengths t1, t2, t3, . . .. At the ith step the man runs for time ti in a straight line that is perpendicular to his radius vector at the start of the step. He chooses to run into the half plane that does not contain the lion (if the lion is on the radius then either direction will do). So certainly the lion does not catch the man in this time step. The man then repeats this procedure for the next time step, and so on.