Infinitely many mathematicians walk into a bar. The first says, “I’ll have a beer.” The second says, “I’ll have half a beer.” The third says, “I’ll have a quarter of a beer.” Before anyone else can speak, the barman fills up exactly two glasses of beer and serves them. “Come on, now,” he says to the group, “You guys have got to learn your limits.”
An engineer, a physicist and a mathematician are staying in a hotel. The engineer wakes up and smells smoke. He goes out into the hallway and sees a fire, so he fills a trashcan from his room with water and douses the fire. He goes back to bed.
Later, the physicist wakes up and smells smoke. He opens his door and sees a fire in the hallway. He walks down the hall to a fire hose and after calculating the flame velocity, distance, water pressure, trajectory, etc. extinguishes the fire with the minimum amount of water and energy needed.
1.Let f : [0, 1] → R satisfy |f(x) − f(y)| ≤6 |x − y| for all x, y ∈ [0, 1]. Show that f is continuous and that for all ε > 0, there exists a piece-wise constant function g such that sup x∈[0,1] |f(x) − g(x)| <6 ε. 2.For all integers n > 1, let un = $$∫^1_0$$f(t) cos(nt)dt. Show that the sequence (un) converges to 0.