A certain lecture has finitely many students and at least two students. Every student fell asleep exactly once and woke up exactly once. Suppose that for any two students, there was some time at which both were asleep. Prove that there was a time at which all the students were asleep.
Solution
Solution:
Let Alice be the first student to wake up. Then, immediately before Alice wakes up, all students must have been asleep, since every other student must have slept at the same time as Alice, and none of them have woken up yet. Thus, all the students are asleep.
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