Monk Puzzle

Here’s an amusing puzzle, from Marginal Revolution:

One morning, exactly at sunrise, a Buddhist monk began to climb a tall
mountain. The narrow path, no more than a foot or two wide, spiraled
around the mountain to a glittering temple at the summit. The monk
ascended the path at varying rates of speed, stopping many times along
the way to rest and to eat the dried fruit he carried with him. He
reached the temple shortly before sunset. After several days of fasting
and meditation he began his journey back along the same path, starting
at sunrise and again walking at variable speeds with many pauses along
the way. His average speed descending was, of course, greater than his
average climbing speed.

Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

My favorite approach to the solution is in the last paragraph.

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