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3) Prove that there exists f : R → R non-negative and continuous such that f...
4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists 8 >0 such that f(x) > 8 for any x € [0,1].
Prove the ratio test . What does this tell you if exists? (Ratio test) If for all sufficiently large n and some r < 1, then converges absolutely; while if for all sufficiently large n, then diverges. lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
4. Prove that the external measure is not "additive", in the sense that [0, 1]le < Vle+ |[0, 1] \Vle. 5. Let 7(x) := lim sup fx(x). +00 Prove that, for every a € R, {7 > a} = UN U{n>a + m}: MENJEN
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
(Advanced Calculus and Real Analysis) - Lebesgue integral, Convergence properties of the Integral for Non-negative functions * Supposef is a nonnegative M-measurable function with Soofd) <0. Then we define the Laplace transform of f, denoted, F, by F(t) = -f(x) dx(r), t> 0. J[0,00) Show that a) F is real valued. b) F is continuous on (0,0). Hint: First establish that F is nonincreasing, c) lim- F(t) = 0.
Please prove this, thanks! 2. Let {xn n21 be a sequence in R such that all n > 0. If ( lim supra) . (lim supー) = 1 Tn (here we already assume both factors are finite), prove that converges.
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
1. Let x, a € R. Prove that if a <a, then -a < x <a.