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, reaching basecamp by nightfall. (Even though the going is easier, he stops a number of times to admire the scenery and rest.)
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?
He need not neither start just at sunup, or arrive back at sunset; as long as he doesn't start back
earlier than the original start, or
later than the original arrival, there must be such a place as asked for.
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