Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R. Let f, (x) := lxl1+1/n, Π ε N, and f(...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but (Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a (5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
(1) Let {fn} < C[a, b], and let {xn} c [a, b]. Suppose that fn + f uniformly on [a, b] and In + x (as n +00. Show that limn7 fn(2n) = f(x). [3] To a + + ( f or intuico te fan a ant [Dannt u
Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
suppose that f is uniformly continuous on fn(x)=f(x+1/n) converges uniformly to f on 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0< p<1, and k and n are positive integers such that k < n. 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0
help me solve 24.11 24.11 Let fn(x) = x and gn(x) = 7 for all x € R. Let f(x) = x and g(x) = 0 for x E R. (a) Observe fn + f uniformly on R (obvious!) and In + g uniformly on R (almost obvious]. (b) Observe the sequence (fron) does not converge uniformly to fg on R. Compare Exercise 24.2. 24.2 For x € [0,00), let fn(x) = (a) Find f(x) = lim fn(x). (b) Determine...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...