A Fixed-Point Theorem 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 presicely the same time of day.