9.K Let X = Q y K = {p € Q : 2 < p? < 3} . Show that K is closed in Q, is bounded, and is not compact in Q.
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- B- use Chebychev's inequality, to obtain an upper bounded on p{|X-u |>2} and compute it with actual value4. a) Let Z suPo 0. Hint: Kolmogorov's inequality. b) Show that lim ta X₂ = 0 t-x P-a.s. for every a > 1/2. Hint: Use a), Aufgabe 3, and the strong law of large numbers. c) Conclude that Y = (Yt)te[0,7] with Y=t X1/t for t> 0 and Yo:= 0 is a Brownian motion.1
- Let A = (0, 1]-{|n € N}. (a) Find the set Aº of its interior points. (b) Find the set A' of its limit points. (c) Determine whether A is open or closed. (Discuss both.) (d) Determine whether A is compact. (e) Determine whether A is connected. (f) Find the set 8A of its boundary points.Let A and B be subsets of R which are bounded above. a. Show that sup(AUB) = max {sup(A), sup(B)}. Let y = sup (A U B), a = sup (A), and ß = sup (B)4. Let X be a metric space. Show that if every continuous function f X → Ris bounded, then X is compact.
- Suppose P is a partition of the interval [a, b], and f: [a, b] → R and g: [a, b] → R are bounded functions. True or false: The upper Darboux sum U(P, f + g) equals U(P, f)+U(P, g). O True O FalseCan you have a class boundary that is less than zero? On the probelem I am working, the class goes from 0-5Q1: Find Sup ; Inf; Max; Min for the following sets: m { e z*} . (a) S = 2n : m,n E Z* }; (b) T = {n+1 :n E Z+ Q2: (a) Let a ,b E R, and a < b. Prove that 3s ER- Q, aRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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