Is there a spot?

A mountain climber is journeying up a mountain trail. He starts out with the sunrise, and gets to the peak just as the sun is setting. Throughout the day, he moves with inconsistent speed, slowing down to rest a few times. He spends the night at the top, and in the morning sets out downward, along the same trail/path he did the day before, reaching basecamp by nightfall.
Is there an altitude on the mountain slope for which the climber was at that exact altitude at the same point during both days? Why or why not?


Answer:

There has to be a spot necessarily upon which he should be at the same time at the same spot on both the days whose altitude must be the same. In order to perceive this fact clearly, just imagine that 2 different persons are simultaneously trodding on the path traversed by the man. One upward and another one downward. They should meet at a spot because they are stepping on the same path. This spot is the required one.