Lee R. answered • 10/07/15

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Mathematics, physics, computer programming

If the position of the monk down the path is Down(t), and the position of the monk up is Up(t) where t is the time of day and the month walks continuously then both Down() and Up() are continuous functions.

The difference between the positions, Diff(t) = Up(t) - Down(t) {measured from the monastery} is also a continuous function.

If L is the length of the path measured from the monastery, then Down(t

_{0}) = 0 and Up(t_{0}) = LSince Up(t

_{0}) > Down(t_{0}) where t_{0}is the start of the respective days, then Diff(t_{0}) = L which is greater than 0.At the end of the day, t

_{f}, Up(t_{f}) = 0, Down(t_{f}) = L and Diff(t_{f}) = 0 - L.Up(t

_{f}) < Down(t_{f}) and Diff(t_{f}) is negative because the monk has finished the journey.Since Diff(t) is continuous, by the intermediate value theorem, if Diff(t

_{0}) > 0 and Diff(t_{f}) < 0, it must pass through a point where Diff(t) = 0. At this point Down(t) = Up(t). At time t the monk is at the same point along the path on both days.