Learning To Reason With Two Monks

I took a training course at work a number of years ago.

The class was on creative thinking, or left brained thinking, or problem solving.

The point was to try to get the students to approach problems in a different manner.

At one point, the instructor posed the following problem, and then moved along with the lesson without providing an answer:

"At dawn, a monk stands on a path at the base of a mountain. The path is only wide enough for one person at a time. As the sun rises, the monk starts moving along the path. Sometimes he walks, sometimes he runs, sometimes he kneels and prays. As the sun sets, he reaches the monastery at the end of the path at the top of the mountain.
The next morning, at dawn, he stands at the top of the same path, but facing down the mountain. As the sun rises, he starts to move down the path. Sometimes he walks, sometimes he runs, sometimes he kneels and prays. As the sun sets, he reaches the same spot where he had started the day before.

Prove that there is a placed on the path that the monk stood in the exact same spot at exactly the same time on both days."