"Proving" a Spot exists

[From "Conceptual Blockbusting" by James Adams]

"It is always of interest to see the variation in thinking among a number of people. Try [this] exercise on your friends and associates ..."

"Puzzle: One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. A narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended at varying rates of speed, stopping many times along the way to rest and eat 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, was greater than his average climbing speed. Prove that here is a spot along the path that the monk will occupy on both trips at precisely the same time of day."