Problem: Let $f: [0,1] \to [0,1]$ be either monotonic increasing. Then prove or disprove that there exists an $x\in [0,1]$ such that $f(x) = x$. Is your answer going to change if we replace monotonic increasing to monotonic decreasing?
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.